Vector Algebra

199 questions · Mathematics · JEE Main

Vector Algebra

199 questions · Mathematics · JEE Main

If $a$ is a nonzero vector such that its projections on the vectors $2 \hat{i} - \hat{j} + 2 \hat{k}, \hat{i} + 2 \hat{j} - 2 \hat{k}$ and $\hat{k}$ are equal, then a unit vector along $a$ is :2025 · 2 Apr · Shift 1 · Q38 · MCQ
Let $a = 2 \hat{i} - 3 \hat{j} + \hat{k}, b = 3 \hat{i} + 2 \hat{j} + 5 \hat{k}$ and a vector $c$ be such that $(a - c) \times b = - 18 \hat{i} - 3 \hat{j} + 12 \hat{k}$ …2025 · 2 Apr · Shift 2 · Q44 · MCQ
Let $a = \hat{i} + \hat{j} + \hat{k}, b = 3 \hat{i} + 2 \hat{j} - \hat{k}, c = λ \hat{j} + μ \hat{k}$ and $\hat{d}$ be a unit vector such that $a \times \hat{d} = b \times \hat{d}$ and $c \cdot \hat{d} = 1$ . If $c$ …2025 · 3 Apr · Shift 1 · Q47 · Numerical
Let $a = \hat{i} + 2 \hat{j} + \hat{k}, b = 3 \hat{i} - 3 \hat{j} + 3 \hat{k}, c = 2 \hat{i} - \hat{j} + 2 \hat{k}$ and $d$ be a vector such that $b \times d = c \times d$ and $a \cdot d = 4$ . Then $∣ (a \times d) ∣^{2}$ …2025 · 3 Apr · Shift 2 · Q49 · Numerical
Consider two vectors $u = 3 \hat{i} - \hat{j}$ and $v = 2 \hat{i} + \hat{j} - λ \hat{k}, λ > 0$ . The angle between them is given by $cos^{- 1} (\frac{5}{2 7})$ . Let $v = v_{1} + v_{2}$ …2025 · 4 Apr · Shift 1 · Q38 · MCQ
Let the three sides of a triangle ABC be given by the vectors $2 \hat{i} - \hat{j} + \hat{k}, \hat{i} - 3 \hat{j} - 5 \hat{k}$ and $3 \hat{i} - 4 \hat{j} - 4 \hat{k}$ . Let $G$ be the centroid of the triangle $A B C$ . Then $6 (∣ AG ∣^{2} + ∣ BG ∣^{2} + ∣ CG ∣^{2})$ …2025 · 4 Apr · Shift 2 · Q48 · Numerical
Let the angle $θ, 0 < θ < \frac{π}{2}$ between two unit vectors $\overset{a}{^}$ and $\hat{b}$ be $sin^{- 1} (\frac{65}{9})$ . If the vector $c = 3 \overset{a}{^} + 6 \hat{b} + 9 (\overset{a}{^} \times \hat{b})$ , then the value of…2025 · 7 Apr · Shift 1 · Q44 · MCQ
Let $a$ and $b$ be the vectors of the same magnitude such that $\frac{∣ a + b ∣ + ∣ a - b ∣}{∣ a + b ∣ - ∣ a - b ∣} = 2 + 1.$ Then $\frac{∣ a + b ∣ ^{2}}{∣ a ∣ ^{2}}$ is :2025 · 7 Apr · Shift 2 · Q41 · MCQ
Let $a = \hat{i} + 2 \hat{j} + \hat{k}$ and $b = 2 \hat{i} + \hat{j} - \hat{k}$ . Let $\overset{c}{^}$ be a unit vector in the plane of the vectors $a$ and $b$ and be perpendicular to $a$ . Then such a vector $\overset{c}{^}$ …2025 · 8 Apr · Shift 2 · Q43 · MCQ
Let $c$ be the projection vector of $b = λ \hat{i} + 4 \hat{k}, λ > 0$ , on the vector $a = \hat{i} + 2 \hat{j} + 2 \hat{k}$ . If $∣ a + c ∣ = 7$ , then the area of the parallelogram formed by the vectors $b$ …2025 · 22 Jan · Shift 1 · Q46 · Numerical
Let $a$ and $b$ be two unit vectors such that the angle between them is $\frac{π}{3}$ . If $λ a + 2 b$ and $3 a - λ b$ are perpendicular to each other, then the number of values of $λ$ in $[- 1, 3]$ …2025 · 22 Jan · Shift 2 · Q36 · MCQ
Let the position vectors of the vertices $A, B$ and C of a tetrahedron ABCD be $\hat{i} + 2 \hat{j} + \hat{k}, \hat{i} + 3 \hat{j} - 2 \hat{k}$ and $2 \hat{i} + \hat{j} - \hat{k}$ …2025 · 23 Jan · Shift 1 · Q30 · MCQ
Let the arc $A C$ of a circle subtend a right angle at the centre $O$ . If the point $B$ on the arc $A C$ , divides the arc $A C$ such that $\frac{length of arc A B}{length of arc B C} = \frac{1}{5}$ …2025 · 23 Jan · Shift 1 · Q42 · MCQ
Let the point A divide the line segment joining the points $P (- 1, - 1, 2)$ and $Q (5, 5, 10)$ internally in the ratio $r : 1 (r > 0)$ . If O is the origin and $(OQ \cdot OA) - \frac{1}{5} ∣ OP \times OA ∣^{2} = 10$ …2025 · 23 Jan · Shift 2 · Q29 · MCQ
Let $a = \hat{i} + 2 \hat{j} + 3 \hat{k}, b = 3 \hat{i} + \hat{j} - \hat{k}$ and $c$ be three vectors such that $c$ is coplanar with $a$ and $b$ . If the vector $C$ is perpendicular to $b$ and $a \cdot c = 5$ …2025 · 24 Jan · Shift 1 · Q38 · MCQ
Let the position vectors of three vertices of a triangle be $4 p + q - 3 r, - 5 p + q + 2 r$ and $2 p - q + 2 r$ . If the position vectors of the orthocenter and the circumcenter of the triangle are $\frac{p + q + r}{4}$ …2025 · 24 Jan · Shift 2 · Q35 · MCQ
Let $a = 3 \hat{i} - \hat{j} + 2 \hat{k}, b = a \times (\hat{i} - 2 \hat{k})$ and $c = b \times \hat{k}$ . Then the…2025 · 24 Jan · Shift 2 · Q38 · MCQ
Let $a = \hat{i} + \hat{j} + \hat{k}, b = 2 \hat{i} + 2 \hat{j} + \hat{k}$ and $d = a \times b$ . If $c$ …2025 · 28 Jan · Shift 1 · Q50 · Numerical
If the components of $a = α \hat{i} + β \hat{j} + γ \hat{k}$ along and perpendicular to $b = 3 \hat{i} + \hat{j} - \hat{k}$ respectively, are $\frac{16}{11} (3 \hat{i} + \hat{j} - \hat{k})$ and $\frac{1}{11} (- 4 \hat{i} - 5 \hat{j} - 17 \hat{k})$ …2025 · 28 Jan · Shift 2 · Q33 · MCQ
Let $A, B, C$ be three points in xy-plane, whose position vector are given by $3 \hat{i} + \hat{j}, \hat{i} + 3 \hat{j}$ and $a \hat{i} + (1 - a) \hat{j}$ respectively with respect to the origin O . If the distance of the point C…2025 · 28 Jan · Shift 2 · Q38 · MCQ
Let $a = \hat{i} + 2 \hat{j} + \hat{k}$ and $b = 2 \hat{i} + 7 \hat{j} + 3 \hat{k}$ . Let $L_{1} : r = (- \hat{i} + 2 \hat{j} + \hat{k}) + λ a, λ \in R$ and $L_{2} : r = (\hat{j} + \hat{k}) + μ b, μ \in R$ …2025 · 29 Jan · Shift 1 · Q39 · MCQ
Let $a = 2 \hat{i} - \hat{j} + 3 \hat{k}, b = 3 \hat{i} - 5 \hat{j} + \hat{k}$ and $c$ be a vector such that $a \times c = a \times b = c \times b$ and $(a + c) \cdot (b + c) = 168$ …2025 · 29 Jan · Shift 1 · Q41 · MCQ
Let $\overset{a}{^}$ be a unit vector perpendicular to the vectors $b = \hat{i} - 2 \hat{j} + 3 \hat{k}$ and $c = 2 \hat{i} + 3 \hat{j} - \hat{k}$ , and $\overset{a}{^}$ makes an angle of $cos^{- 1} (- \frac{1}{3})$ with the vector $\hat{i} + \hat{j} + \hat{k}$ …2025 · 29 Jan · Shift 2 · Q30 · MCQ
Let $a = - 5 \hat{i} + \hat{j} - 3 \hat{k}, b = \hat{i} + 2 \hat{j} - 4 \hat{k}$ and $c = (((a \times b) \times \hat{i}) \times \hat{i}) \times \hat{i}$ …2024 · 1 Feb · Shift 1 · Q38 · MCQ
Let $a = \hat{i} + \hat{j} + \hat{k}, b = - \hat{i} - 8 \hat{j} + 2 \hat{k}$ and $c = 4 \hat{i} + c_{2} \hat{j} + c_{3} \hat{k}$ be three vectors such that $b \times a = c \times a$ …2024 · 1 Feb · Shift 2 · Q58 · Numerical
Let a unit vector which makes an angle of $6 0^{\circ}$ with $2 \hat{i} + 2 \hat{j} - \hat{k}$ and an angle of $4 5^{\circ}$ with $\hat{i} - \hat{k}$ be $C$ . Then $C + (- \frac{1}{2} \hat{i} + \frac{1}{3 2} \hat{j} - \frac{2}{3} \hat{k})$ …2024 · 4 Apr · Shift 1 · Q38 · MCQ
Let $ABC$ be a triangle of area $152$ and the vectors $AB = \hat{i} + 2 \hat{j} - 7 \hat{k}, BC = a \hat{i} + b \hat{j} + c \hat{k}$ and $AC = 6 \hat{i} + d \hat{j} - 2 \hat{k}, d > 0$ …2024 · 4 Apr · Shift 1 · Q59 · Numerical
For $λ > 0$ , let $θ$ be the angle between the vectors $a = \hat{i} + λ \hat{j} - 3 \hat{k}$ and $b = 3 \hat{i} - \hat{j} + 2 \hat{k}$ . If the vectors $a + b$ and $a - b$ are mutually perpendicular,…2024 · 4 Apr · Shift 2 · Q40 · MCQ
Let $a = \hat{i} + \hat{j} + \hat{k}, b = 2 \hat{i} + 4 \hat{j} - 5 \hat{k}$ and $c = x \hat{i} + 2 \hat{j} + 3 \hat{k}, x \in R$ . If $d$ is the unit vector in the direction of $b + c$ such that $a \cdot d = 1$ …2024 · 4 Apr · Shift 2 · Q50 · MCQ
If $A (1, - 1, 2), B (5, 7, - 6), C (3, 4, - 10)$ and $D (- 1, - 4, - 2)$ are the vertices of a quadrilateral ABCD, then its area is :2024 · 5 Apr · Shift 1 · Q49 · MCQ
Let $a = \hat{i} - 3 \hat{j} + 7 \hat{k}, b = 2 \hat{i} - \hat{j} + \hat{k}$ and $c$ be a vector such that $(a + 2 b) \times c = 3 (c \times a)$ …2024 · 5 Apr · Shift 1 · Q58 · Numerical
Let $a = 2 \hat{i} + 5 \hat{j} - \hat{k}, b = 2 \hat{i} - 2 \hat{j} + 2 \hat{k}$ and $c$ be three vectors such that $(c + \hat{i}) \times (a + b + \hat{i}) = a \times (c + \hat{i})$ . If $a \cdot c = - 29$ …2024 · 5 Apr · Shift 2 · Q37 · MCQ
Consider three vectors $a, b, c$ . Let $∣ a ∣ = 2, ∣ b ∣ = 3$ and $a = b \times c$ . If $α \in [0, \frac{π}{3}]$ is the angle between the vectors $b$ and $c$ , then the…2024 · 5 Apr · Shift 2 · Q49 · MCQ
Let $a = 2 \hat{i} - 3 \hat{j} + 4 \hat{k}, b = 3 \hat{i} + 4 \hat{j} - 5 \hat{k}$ and a vector $c$ be such that $a \times (b + c) + b \times c = \hat{i} + 8 \hat{j} + 13 \hat{k}$ . If $\vec{a} \cdot…2024 · 6 Apr · Shift 1 · Q59 · Numerical
Let $a = 2 \hat{i} + \hat{j} - \hat{k}, b = ((a \times (\hat{i} + \hat{j})) \times \hat{i}) \times \hat{i}$ . Then the square of the projection of $a$ on $b$ is:2024 · 6 Apr · Shift 2 · Q37 · MCQ
Let $a = 6 \hat{i} + \hat{j} - \hat{k}$ and $b = \hat{i} + \hat{j}$ . If $c$ is a is vector such that $∣ c ∣ \geq 6, a \cdot c = 6∣ c ∣, ∣ c - a ∣ = 22$ …2024 · 6 Apr · Shift 2 · Q38 · MCQ
The set of all $α$ , for which the vectors $a = α t \hat{i} + 6 \hat{j} - 3 \hat{k}$ and $b = t \hat{i} - 2 \hat{j} - 2 α t \hat{k}$ are inclined at an obtuse angle for all $t \in R$ , is2024 · 8 Apr · Shift 1 · Q34 · MCQ
Let $a = 9 \hat{i} - 13 \hat{j} + 25 \hat{k}, b = 3 \hat{i} + 7 \hat{j} - 13 \hat{k}$ and $c = 17 \hat{i} - 2 \hat{j} + \hat{k}$ be three given vectors. If $r$ is a vector such that $r \times a = (b + c) \times a$ …2024 · 8 Apr · Shift 1 · Q53 · Numerical
Let $a = 4 \hat{i} - \hat{j} + \hat{k}, b = 11 \hat{i} - \hat{j} + \hat{k}$ and $c$ be a vector such that $(a + b) \times c = c \times (- 2 a + 3 b)$ …2024 · 8 Apr · Shift 2 · Q35 · MCQ
Let $a = \hat{i} + 2 \hat{j} + 3 \hat{k}, b = 2 \hat{i} + 3 \hat{j} - 5 \hat{k}$ and $c = 3 \hat{i} - \hat{j} + λ \hat{k}$ be three vectors. Let $r$ …2024 · 8 Apr · Shift 2 · Q44 · MCQ
Let three vectors , $a = α \hat{i} + 4 \hat{j} + 2 \hat{k}, b = 5 \hat{i} + 3 \hat{j} + 4 \hat{k}, c = x \hat{i} + y \hat{j} + z \hat{k}$ form a triangle such that $c = a - b$ …2024 · 9 Apr · Shift 1 · Q32 · MCQ
Let $O A = 2 a, O B = 6 a + 5 b$ and $O C = 3 b$ , where $O$ is the origin. If the area of the parallelogram with adjacent sides $O A$ and $O C$ …2024 · 9 Apr · Shift 1 · Q50 · MCQ
Between the following two statements: Statement I : Let $a = \hat{i} + 2 \hat{j} - 3 \hat{k}$ and $b = 2 \hat{i} + \hat{j} - \hat{k}$ . Then the vector $r$ satisfying $a \times r = a \times b$ and $a \cdot r = 0$ …2024 · 9 Apr · Shift 2 · Q42 · MCQ
Let $a = 2 \hat{i} + α \hat{j} + \hat{k}, b = - \hat{i} + \hat{k}, c = β \hat{j} - \hat{k}$ , where $α$ and $β$ are integers and $α β = - 6$ . Let the values of the ordered pair $(α, β)$ , for which the…2024 · 9 Apr · Shift 2 · Q50 · MCQ
Let $a = \hat{i} + 2 \hat{j} + \hat{k}, b = 3 (\hat{i} - \hat{j} + \hat{k})$ . Let $c$ be the vector such that $a \times c = b$ …2024 · 27 Jan · Shift 1 · Q48 · MCQ
The least positive integral value of $α$ , for which the angle between the vectors $α \hat{i} - 2 \hat{j} + 2 \hat{k}$ and $α \hat{i} + 2 α \hat{j} - 2 \hat{k}$ is acute, is .2024 · 27 Jan · Shift 1 · Q58 · Numerical
Let the position vectors of the vertices $A, B$ and $C$ of a triangle be $2 \hat{i} + 2 \hat{j} + \hat{k}, \hat{i} + 2 \hat{j} + 2 \hat{k}$ and $2 \hat{i} + \hat{j} + 2 \hat{k}$ respectively. Let $l_{1}, l_{2}$ and $l_{3}$ be…2024 · 27 Jan · Shift 2 · Q32 · MCQ
The position vectors of the vertices $A, B$ and $C$ of a triangle are $2 \hat{i} - 3 \hat{j} + 3 \hat{k}, 2 \hat{i} + 2 \hat{j} + 3 \hat{k}$ and $- \hat{i} + \hat{j} + 3 \hat{k}$ respectively. Let $l$ denotes the length of…2024 · 27 Jan · Shift 2 · Q50 · MCQ
Let $a, b$ and $c$ be three non-zero vectors such that $b$ and $c$ are non-collinear. If $a + 5 b$ is collinear with $c, b + 6 c$ is collinear with $a$ and $a + α b + β c = 0$ …2024 · 29 Jan · Shift 1 · Q32 · MCQ
Let a unit vector $\overset{u}{^} = x \hat{i} + y \hat{j} + z \hat{k}$ make angles $\frac{π}{2}, \frac{π}{3}$ and $\frac{2 π}{3}$ with the vectors $\frac{1}{2} \hat{i} + \frac{1}{2} \hat{k}, \frac{1}{2} \hat{j} + \frac{1}{2} \hat{k}$ …2024 · 29 Jan · Shift 2 · Q35 · MCQ
Let $O A = a, O B = 12 a + 4 b and O C = b$ , where O is the origin. If S is the parallelogram with adjacent sides OA and OC, then $\frac{area of the quadrilateral OA BC}{area of S}$ …2024 · 29 Jan · Shift 2 · Q44 · MCQ
Let $a = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}$ and $b = b_{1} \hat{i} + b_{2} \hat{j} + b_{3} \hat{k}$ be two vectors such that $∣ a ∣ = 1, a \cdot b = 2$ …2024 · 30 Jan · Shift 1 · Q34 · MCQ
Let $a = \hat{i} + α \hat{j} + β \hat{k}, α, β \in R$ . Let a vector $b$ be such that the angle between $a$ and $b$ is $\frac{π}{4}$ and $∣ b ∣^{2} = 6$ . If $\vec{a} \cdot \vec{b}=3…2024 · 30 Jan · Shift 2 · Q42 · MCQ
Let $a$ and $b$ be two vectors such that $∣ b ∣ = 1$ and $∣ b \times a ∣ = 2$ . Then $∣ (b \times a) - b ∣^{2}$ is equal to2024 · 30 Jan · Shift 2 · Q47 · MCQ
Let $a = 3 \hat{i} + \hat{j} - 2 \hat{k}, b = 4 \hat{i} + \hat{j} + 7 \hat{k}$ and $c = \hat{i} - 3 \hat{j} + 4 \hat{k}$ be three vectors. If a vectors $p$ satisfies $p \times b = c \times b$ and $p \cdot a = 0$ …2024 · 31 Jan · Shift 1 · Q39 · MCQ
The distance of the point $Q (0, 2, - 2)$ form the line passing through the point $P (5, - 4, 3)$ and perpendicular to the lines $r = (- 3 \hat{i} + 2 \hat{k}) + λ (2 \hat{i} + 3 \hat{j} + 5 \hat{k}), λ \in R$ and $r = (\hat{i} - 2 \hat{j} + \hat{k}) + μ (- \hat{i} + 3 \hat{j} + 2 \hat{k}), μ \in R$ …2024 · 31 Jan · Shift 1 · Q42 · MCQ
Let $a$ and $b$ be two vectors such that $∣ a ∣ = 1, ∣ b ∣ = 4$ , and $a \cdot b = 2$ . If $c = (2 a \times b) - 3 b$ and the angle between $b$ and $c$ is $α$ , then $192 sin^{2} α$ …2024 · 31 Jan · Shift 1 · Q53 · Numerical
Let $a = 3 \hat{i} + 2 \hat{j} + \hat{k}, b = 2 \hat{i} - \hat{j} + 3 \hat{k}$ and $c$ be a vector such that $(a + b) \times c = 2 (a \times b) + 24 \hat{j} - 6 \hat{k}$ and $(a - b + \hat{i}) \cdot c = - 3$ …2024 · 31 Jan · Shift 2 · Q51 · Numerical
$A (2, 6, 2), B (- 4, 0, λ), C (2, 3, - 1)$ and $D (4, 5, 0), ∣ λ ∣ \leq 5$ are the vertices of a quadrilateral $A B C D$ . If its area is 18 square units, then $5 - 6 λ$ is equal to .2023 · 1 Feb · Shift 1 · Q42 · Numerical
Let $a = 5 \hat{i} - \hat{j} - 3 \hat{k}$ and $b = \hat{i} + 3 \hat{j} + 5 \hat{k}$ be two vectors. Then which one of the following statements is TRUE ?2023 · 1 Feb · Shift 2 · Q27 · MCQ
Let $a = 2 \hat{i} - 7 \hat{j} + 5 \hat{k}, b = \hat{i} + \hat{k}$ and $c = \hat{i} + 2 \hat{j} - 3 \hat{k}$ be three given vectors. If $r$ is a vector such that $r \times a = c \times a$ …2023 · 1 Feb · Shift 2 · Q36 · MCQ
Let $a = 2 \hat{i} + 3 \hat{j} + 4 \hat{k}, b = \hat{i} - 2 \hat{j} - 2 \hat{k}$ and $c = - \hat{i} + 4 \hat{j} + 3 \hat{k}$ . If $d$ is a vector perpendicular to both $b$ and $c$ , and $a \cdot d = 18$ , then $∣ a \times d ∣^{2}$ …2023 · 6 Apr · Shift 1 · Q31 · MCQ
If the points with position vectors $α \hat{i} + 10 \hat{j} + 13 \hat{k}, 6 \hat{i} + 11 \hat{j} + 11 \hat{k}, \frac{9}{2} \hat{i} + β \hat{j} - 8 \hat{k}$ are collinear, then $(19 α - 6 β)^{2}$ is equal to :2023 · 8 Apr · Shift 1 · Q38 · MCQ
Let $a = 6 \hat{i} + 9 \hat{j} + 12 \hat{k}, b = α \hat{i} + 11 \hat{j} - 2 \hat{k}$ and $c$ be vectors such that $a \times c = a \times b$ . If $a \cdot c = - 12, c \cdot (\hat{i} - 2 \hat{j} + \hat{k}) = 5$ …2023 · 8 Apr · Shift 1 · Q43 · Numerical
The area of the quadrilateral $ABCD$ with vertices $A (2, 1, 1), B (1, 2, 5), C (- 2, - 3, 5)$ and $D (1, - 6, - 7)$ is equal to :2023 · 8 Apr · Shift 2 · Q29 · MCQ
Let O be the origin and the position vector of the point P be $- i - 2 j + 3 k$ . If the position vectors of the points A, B and C are $- 2 i + j - 3 k, 2 i + 4 j - 2 k$ …2023 · 10 Apr · Shift 1 · Q27 · MCQ
An arc PQ of a circle subtends a right angle at its centre O. The mid point of the arc PQ is R. If $O P = u, O R = v$ , and $O Q = α u + β v$ …2023 · 10 Apr · Shift 1 · Q30 · MCQ
Let $a = 2 \hat{i} + 7 \hat{j} - \hat{k}, b = 3 \hat{i} + 5 \hat{k}$ and $c = \hat{i} - \hat{j} + 2 \hat{k}$ . Let $d$ be a vector which is perpendicular to both $a$ and $b$ , and $c \cdot d = 12$ . Then $(- \hat{i} + \hat{j} - \hat{k}) \cdot (c \times d)$ …2023 · 10 Apr · Shift 2 · Q29 · MCQ
If the points $P$ and $Q$ are respectively the circumcenter and the orthocentre of a $△ ABC$ , then $PA + PB + PC$ is equal to :2023 · 10 Apr · Shift 2 · Q30 · MCQ
For any vector $a = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}$ , with $10 ∣ a_{i} ∣ (A) :$ \max \left\{\left|a_{1}\right|,\left|a_{2}\right|,\left|a_{3}\right|\right\} \leq|\vec{a}| $(B) :$ |\vec{a}| \leq 3 \max…2023 · 11 Apr · Shift 1 · Q32 · MCQ
Let $a$ be a non-zero vector parallel to the line of intersection of the two planes described by $\hat{i} + \hat{j}, \hat{i} + \hat{k}$ and $\hat{i} - \hat{j}, \hat{j} - \hat{k}$ . If $θ$ is the angle between the vector $a$ and the…2023 · 11 Apr · Shift 1 · Q37 · MCQ
Let $a = \hat{i} + 2 \hat{j} + 3 \hat{k}$ and $b = \hat{i} + \hat{j} - \hat{k}$ . If $c$ is a vector such that $a \cdot c = 11, b \cdot (a \times c) = 27$ and $b \cdot c = - 3 ∣ b ∣$ , then…2023 · 11 Apr · Shift 2 · Q37 · Numerical
Let $a = \hat{i} + 4 \hat{j} + 2 \hat{k}, b = 3 \hat{i} - 2 \hat{j} + 7 \hat{k}$ and $c = 2 \hat{i} - \hat{j} + 4 \hat{k}$ . If a vector $d$ satisfies $d \times b = c \times b$ and $d \cdot a = 24$ ,…2023 · 13 Apr · Shift 1 · Q25 · MCQ
Let $a = 3 \hat{i} + \hat{j} - \hat{k}$ and $c = 2 \hat{i} - 3 \hat{j} + 3 \hat{k}$ . If $b$ is a vector such that $a = b \times c$ and $∣ b ∣^{2} = 50$ , then $∣72 - ∣ b + c ∣^{2} ∣$ is equal…2023 · 13 Apr · Shift 1 · Q43 · Numerical
Let $∣ a ∣ = 2, ∣ b ∣ = 3$ and the angle between the vectors $a$ and $b$ be $\frac{π}{4}$ . Then $∣ (a + 2 b) \times (2 a - 3 b) ∣^{2}$ is equal to :2023 · 13 Apr · Shift 2 · Q22 · MCQ
Let for a triangle $ABC$ , $AB = - 2 \hat{i} + \hat{j} + 3 \hat{k} CB = α \hat{i} + β \hat{j} + γ \hat{k} CA = 4 \hat{i} + 3 \hat{j} + δ \hat{k}$ If $δ > 0$ …2023 · 13 Apr · Shift 2 · Q34 · MCQ
Let $ABCD$ be a quadrilateral. If $E$ and $F$ are the mid points of the diagonals $AC$ and $BD$ respectively and $(A B - B C) + (A D - D C) = k F E$ …2023 · 15 Apr · Shift 1 · Q27 · MCQ
Let PQR be a triangle. The points A, B and C are on the sides QR, RP and PQ respectively such that $\frac{Q A}{A R} = \frac{R B}{B P} = \frac{P C}{C Q} = \frac{1}{2}$ . Then $\frac{A r e a ( Δ P QR )}{A r e a ( Δ A B C )}$ is equal…2023 · 24 Jan · Shift 1 · Q35 · MCQ
Let $α = 4 i + 3 j + 5 k$ and $β = i + 2 j - 4 k$ . Let $β_{1}$ be parallel to $α$ and $β_{2}$ …2023 · 24 Jan · Shift 2 · Q26 · MCQ
Let $a = i + 2 j + λ k, b = 3 i - 5 j - λ k, a . c = 7, 2 b . c + 43 = 0, a \times c = b \times c$ …2023 · 24 Jan · Shift 2 · Q39 · Numerical
The vector $a = - i + 2 j + k$ is rotated through a right angle, passing through the y-axis in its way and the resulting vector is $b$ . Then the projection of $3 a + 2 b$ …2023 · 25 Jan · Shift 1 · Q24 · MCQ
If the vectors $a = λ i + μ j + 4 k$ , $b = - 2 i + 4 j - 2 k$ and $c = 2 i + 3 j + k$ are coplanar and…2023 · 29 Jan · Shift 1 · Q39 · MCQ
If $a = i + 2 k, b = i + j + k, c = 7 i - 3 j + 4 k, r \times b + b \times c = 0$ …2023 · 29 Jan · Shift 2 · Q27 · MCQ
Let $a = 4 i + 3 j$ and $b = 3 i - 4 j + 5 k$ . If $c$ is a vector such that $c . (a \times b) + 25 = 0, c . (i + j + k) = 4$ …2023 · 29 Jan · Shift 2 · Q33 · MCQ
Let a unit vector $O P$ make angles $α, β, γ$ with the positive directions of the co-ordinate axes $OX$ , $OY, OZ$ respectively, where $β \in (0, \frac{π}{2})$ . If $OP$ …2023 · 30 Jan · Shift 1 · Q31 · MCQ
Let $a$ and $b$ be two vectors, Let $∣ a ∣ = 1, ∣ b ∣ = 4$ and $a \cdot b = 2$ . If $c = (2 a \times b) - 3 b$ , then the value of $b \cdot c$ is :2023 · 30 Jan · Shift 2 · Q32 · MCQ
Let $a = 2 \hat{i} + \hat{j} + \hat{k}$ , and $b$ and $c$ be two nonzero vectors such that $∣ a + b + c ∣ = ∣ a + b - c ∣$ and $b \cdot c = 0$ . Consider the following two statements: (A) $∣ a + λ c ∣ \geq ∣ a ∣$ …2023 · 31 Jan · Shift 1 · Q35 · MCQ
Let $a$ and $b$ be two vectors such that $∣ a ∣ = 14, ∣ b ∣ = 6$ and $∣ a \times b ∣ = 48$ . Then $(a \cdot b)^{2}$ is equal to .2023 · 31 Jan · Shift 1 · Q45 · Numerical
Let $a = \hat{i} + 2 \hat{j} + 3 \hat{k}, b = \hat{i} - \hat{j} + 2 \hat{k}$ and $c = 5 \hat{i} - 3 \hat{j} + 3 \hat{k}$ be three vectors. If $r$ is a vector such that, $r \times b = c \times b$ and $r \cdot a = 0$ …2023 · 31 Jan · Shift 2 · Q23 · MCQ
Let $a, b, c$ be three vectors such that $∣ a ∣ = 31, 4∣ b ∣ = ∣ c ∣ = 2$ and $2 (a \times b) = 3 (c \times a)$ . If the angle between $b$ and $c$ is $\frac{2 π}{3}$ , then $(\frac{a \times c}{a \cdot b})^{2}$ …2023 · 31 Jan · Shift 2 · Q42 · Numerical
Let $a$ , $b$ be unit vectors. If $c$ be a vector such that the angle between $a$ and $c$ is $\frac{π}{12}$ , and $b = c + 2 (c \times a)$ …2022 · 24 Jun · Shift 1 · Q33 · MCQ
Let $a$ and $b$ be two unit vectors such that $∣ (a + b) + 2 (a \times b) ∣ = 2$ . If $θ \in$ (0, $π$ ) is the angle between $a$ and $b$ , then among the statements :…2022 · 24 Jun · Shift 2 · Q34 · MCQ
Let $ABC$ be a triangle such that $BC = a, CA = b, AB = c, ∣ a ∣ = 62, ∣ b ∣ = 23$ …2022 · 25 Jul · Shift 1 · Q34 · MCQ
Let $a = \hat{i} - \hat{j} + 2 \hat{k}$ and let $b$ be a vector such that $a \times b = 2 \hat{i} - \hat{k}$ and $a \cdot b = 3$ . Then the projection of $b$ on the vector $a - b$ is :2022 · 25 Jul · Shift 2 · Q34 · MCQ
Let $a = a_{1} i + a_{2} j + a_{3} k$ $a_{i} > 0$ , $i = 1, 2, 3$ be a vector which makes equal angles with the coordinate axes OX, OY and OZ. Also, let the projection of $a$ on the…2022 · 25 Jun · Shift 1 · Q34 · MCQ
Let $θ$ be the angle between the vectors $a$ and $b$ , where $∣ a ∣ = 4, ∣ b ∣ = 3$ and $θ \in (\frac{π}{4}, \frac{π}{3})$ . Then $(a - b) \times (a + b)^{2} + 4 (a . b)^{2}$ …2022 · 25 Jun · Shift 1 · Q40 · Numerical
Let $b = i + j + λ k$ , $λ \in$ R. If $a$ is a vector such that $a \times b = 13 i - j - 4 k$ and $a . b + 21 = 0$ …2022 · 25 Jun · Shift 2 · Q42 · Numerical
Let $a = α \hat{i} + \hat{j} - \hat{k}$ and $b = 2 \hat{i} + \hat{j} - α \hat{k}, α > 0$ . If the projection of $a \times b$ …2022 · 26 Jul · Shift 1 · Q36 · MCQ
Let $a = α \hat{i} + \hat{j} + β \hat{k}$ and $b = 3 \hat{i} - 5 \hat{j} + 4 \hat{k}$ be two vectors, such that $a \times b = - \hat{i} + 9 \hat{j} + 12 \hat{k}$ . Then the projection of $b - 2 a$ on $b + a$ …2022 · 27 Jul · Shift 1 · Q34 · MCQ
Let $a$ , $b$ , $c$ be three non-coplanar vectors such that $a \times b$ = 4 $c$ , $b \times c$ = 9 $a$ …2022 · 27 Jul · Shift 2 · Q40 · Numerical
Let $a = i + j - k$ and $c = 2 i - 3 j + 2 k$ . Then the number of vectors $b$ such that $b \times c = a$ …2022 · 27 Jun · Shift 1 · Q32 · MCQ
Let $a$ and $b$ be the vectors along the diagonals of a parallelogram having area $22$ . Let the angle between $a$ and $b$ be acute, $∣ a ∣ = 1$ , and $∣ a . b ∣ = ∣ a \times b ∣$ …2022 · 27 Jun · Shift 2 · Q32 · MCQ
Let the vectors $a = (1 + t) \hat{i} + (1 - t) \hat{j} + \hat{k}, b = (1 - t) \hat{i} + (1 + t) \hat{j} + 2 \hat{k}$ and $c = t \hat{i} - t \hat{j} + \hat{k}, t \in R$ be such that for $α, β, γ \in R, α a + β b + γ c = 0 \Rightarrow α = β = γ = 0$ …2022 · 28 Jul · Shift 1 · Q25 · MCQ
Let a vector $a$ has magnitude 9. Let a vector $b$ be such that for every $(x, y) \in R \times R - {(0, 0)}$ , the vector $(x a + y b)$ is perpendicular to the vector $(6 y a - 18 x b)$ . Then…2022 · 28 Jul · Shift 1 · Q27 · MCQ
Let S be the set of all a $\in R$ for which the angle between the vectors $u = a (lo g_{e} b) \hat{i} - 6 \hat{j} + 3 \hat{k}$ and $v = (lo g_{e} b) \hat{i} + 2 \hat{j} + 2 a (lo g_{e} b) \hat{k}$ , $(b > 1)$ …2022 · 28 Jul · Shift 2 · Q33 · MCQ
If $a = 2 i + j + 3 k$ , $b = 3 i + 3 j + k$ and $c = c_{1} i + c_{2} j + c_{3} k$ are coplanar vectors and $a . c = 5$ …2022 · 28 Jun · Shift 1 · Q38 · Numerical
Let $a = α i + 2 j - k$ and $b = - 2 i + α j + k$ , where $α \in R$ . If the area of the parallelogram whose adjacent sides are…2022 · 28 Jun · Shift 2 · Q35 · MCQ
Let $a$ be a vector which is perpendicular to the vector $3 i + \frac{1}{2} j + 2 k$ . If $a \times (2 i + k) = 2 i - 13 j - 4 k$ …2022 · 28 Jun · Shift 2 · Q39 · MCQ
Let $\overset{a}{^}$ and $\hat{b}$ be two unit vectors such that the angle between them is $\frac{π}{4}$ . If $θ$ is the angle between the vectors $(\overset{a}{^} + \hat{b})$ and $(\overset{a}{^} + 2 \hat{b} + 2 (\overset{a}{^} \times \hat{b}))$ , then the value of $164 cos^{2} θ$ …2022 · 29 Jul · Shift 1 · Q34 · MCQ
Let $a, b, c$ be three coplanar concurrent vectors such that angles between any two of them is same. If the product of their magnitudes is 14 and $(a \times b) \cdot (b \times c) + (b \times c) \cdot (c \times a) + (c \times a) \cdot (a \times b) = 168$ …2022 · 29 Jul · Shift 2 · Q35 · MCQ
Let $a$ and $b$ be two vectors such that $∣ a + b ∣^{2} = ∣ a ∣^{2} + 2∣ b ∣^{2}, a \cdot b = 3$ and $∣ a \times b ∣^{2} = 75$ . Then $∣ a ∣^{2}$ is equal to .2022 · 29 Jul · Shift 2 · Q42 · Numerical
Let $a = α i + 3 j - k$ , $b = 3 i - β j + 4 k$ and $c = i + 2 j - 2 k$ where $\alpha ,\,\beta \in…2022 · 29 Jun · Shift 1 · Q23 · MCQ
Let A, B, C be three points whose position vectors respectively are $a = i + 4 j + 3 k b = 2 i + α j + 4 k, α \in R c = 3 i - 2 j + 5 k$ …2022 · 29 Jun · Shift 2 · Q32 · MCQ
Let $a = i - 2 j + 3 k$ , $b = i + j + k$ and $c$ be a vector such that $a + (b \times c) = 0$ …2022 · 29 Jun · Shift 2 · Q35 · Numerical
Let $a = 2 i - j + 2 k$ and $b = i + 2 j - k$ . Let a vector $v$ be in the plane containing $a$ and $b$ .…2021 · 1 Sep · Shift 2 · Q44 · Numerical
Let a vector $α i + β j$ be obtained by rotating the vector $3 i + j$ by an angle 45 $^{\circ}$ about the origin in counterclockwise direction in the first quadrant. Then the area of…2021 · 16 Mar · Shift 1 · Q23 · MCQ
Let $a$ = $i$ + 2 $j -$ 3 $k$ and $b = 2 i -$ 3 $j$ + 5 $k$ . If $r \times a$ = $b \times r$ , $r$ …2021 · 16 Mar · Shift 2 · Q26 · MCQ
Let O be the origin. Let $O P = x i + y j - k$ and $O Q = - i + 2 j + 3 x k$ , x, y $\in$ R, x > 0, be such that $P Q = 20$ …2021 · 17 Mar · Shift 2 · Q27 · MCQ
Let $x$ be a vector in the plane containing vectors $a = 2 i - j + k$ and $b = i + 2 j - k$ . If the vector $x$ is…2021 · 17 Mar · Shift 2 · Q37 · Numerical
A vector $a$ has components 3p and 1 with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to new system, $a$ …2021 · 18 Mar · Shift 1 · Q31 · MCQ
Let $a$ and $b$ be two non-zero vectors perpendicular to each other and $∣ a ∣ = ∣ b ∣$ . If $∣ a \times b ∣ = ∣ a ∣$ , then the…2021 · 18 Mar · Shift 2 · Q27 · MCQ
In a triangle ABC, if $∣ B C ∣ = 8, ∣ C A ∣ = 7, ∣ A B ∣ = 10$ , then the projection of the vector $A B$ on $A C$ is equal to :2021 · 18 Mar · Shift 2 · Q32 · MCQ
Let $a$ , $b$ , $c$ be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle $θ$ , with the vector $a$ + $b$ + $c$ …2021 · 20 Jul · Shift 1 · Q37 · Numerical
If the shortest distance between the lines $r_{1} = α i + 2 j + 2 k + λ (i - 2 j + 2 k)$ , $λ \in$ R, $α$ > 0 and $r_{2} = - 4 i - k + μ (3 i - 2 j - 2 k)$ …2021 · 20 Jul · Shift 1 · Q40 · Numerical
In a triangle ABC, if $B C = 3$ , $C A = 5$ and $B A = 7$ , then the projection of the vector $B A$ on $B C$ …2021 · 20 Jul · Shift 2 · Q38 · MCQ
For p > 0, a vector $v_{2} = 2 i + (p + 1) j$ is obtained by rotating the vector $v_{1} = 3 p i + j$ by an angle $θ$ about origin in counter clockwise…2021 · 20 Jul · Shift 2 · Q41 · Numerical
Let $a = i + 2 j - k$ , $b = i - j$ and $c = i - j - k$ be three given vectors. If $r$ is a vector…2021 · 25 Feb · Shift 1 · Q44 · Numerical
Let $a = i + α j + 3 k$ and $b = 3 i - α j + k$ . If the area of the parallelogram whose adjacent sides are represented by the vectors $a$ …2021 · 25 Feb · Shift 2 · Q46 · Numerical
Let $p = 2 i + 3 j + k$ and $q = i + 2 j + k$ be two vectors. If a vector $r = (α i + β j + γ k)$ …2021 · 25 Jul · Shift 1 · Q40 · Numerical
Let a, b and c be distinct positive numbers. If the vectors $a i + a j + c k, i + k$ and $c i + c j + b k$ are co-planar, then c is equal to :2021 · 25 Jul · Shift 2 · Q31 · MCQ
If $a = 2, b = 5$ and $a \times b$ = 8, then $a . b$ is equal to :2021 · 25 Jul · Shift 2 · Q34 · MCQ
If $(a + 3 b)$ is perpendicular to $(7 a - 5 b)$ and $(a - 4 b)$ is perpendicular to $(7 a - 2 b)$ …2021 · 25 Jul · Shift 2 · Q43 · Numerical
A hall has a square floor of dimension 10 m $\times$ 10 m (see the figure) and vertical walls. If the angle GPH between the diagonals AG and BH is $cos^{- 1} \frac{1}{5}$ , then the height of the hall (in meters) is : Includes diagram2021 · 26 Aug · Shift 2 · Q35 · MCQ
If the projection of the vector $i + 2 j + k$ on the sum of the two vectors $2 i + 4 j - 5 k$ and $- λ i + 2 j + 3 k$ is 1, then $λ$ is equal to …2021 · 26 Aug · Shift 2 · Q41 · Numerical
If vectors $a_{1} = x i - j + k$ and $a_{2} = i + y j + z k$ are collinear, then a possible unit vector parallel to the vector $x i + y j + z k$ …2021 · 26 Feb · Shift 2 · Q38 · MCQ
Let $a = i + 5 j + α k$ , $b = i + 3 j + β k$ and $c = - i + 2 j - 3 k$ be three vectors such that, $b \times c = 53$ …2021 · 27 Aug · Shift 1 · Q34 · Numerical
Let $a = i + j + 2 k$ and $b = - i + 2 j + 3 k$ . Then the vector product $(a + b) \times ((a \times ((a - b) \times b)) \times b)$ …2021 · 27 Jul · Shift 1 · Q24 · MCQ
Let $a = i + j + k, b$ and $c = j - k$ be three vectors such that $a \times b = c$ and $a . b = 1$ …2021 · 27 Jul · Shift 1 · Q40 · Numerical
Let $a$ and $b$ be two vectors such that $2 a + 3 b = 3 a + b$ and the angle between $a$ and…2021 · 31 Aug · Shift 1 · Q24 · MCQ
Let $a$ , $b$ and $c$ be three unit vectors such that $a - b^{2}$ + $a - c^{2}$ = 8. Then $a + 2 b^{2}$ …2020 · 2 Sep · Shift 1 · Q29 · Numerical
Let the position vectors of points 'A' and 'B' be $i + j + k$ and $2 i + j + 3 k$ , respectively. A point 'P' divides the line segment AB internally in the ratio $λ$ : 1…2020 · 2 Sep · Shift 2 · Q31 · Numerical
The lines $r = (i - j) + l (2 i + k)$ and $r = (2 i - j) + m (i + j + k)$ …2020 · 3 Sep · Shift 1 · Q35 · MCQ
Let a, b c $\in$ R be such that a2 + b2 + c2 = 1. If $a cos θ = b cos (θ + \frac{2 π}{3}) = c cos (θ + \frac{4 π}{3})$ , where $θ = \frac{π}{9}$ , then the angle between the…2020 · 3 Sep · Shift 2 · Q23 · MCQ
If $a = 2 i + j + 2 k$ , then the value of $i \times (a \times i)^{2} + j \times (a \times j)^{2} + k \times (a \times k)^{2}$ …2020 · 4 Sep · Shift 2 · Q33 · Numerical
Let the vectors $a$ , $b$ , $c$ be such that $a = 2$ , $b = 4$ and $c = 4$ . If the…2020 · 5 Sep · Shift 2 · Q26 · Numerical
If $a$ and $b$ are unit vectors, then the greatest value of $3 a + b + a - b$ is …2020 · 6 Sep · Shift 1 · Q27 · Numerical
If $x$ and $y$ be two non-zero vectors such that $x + y = x$ and $2 x + λ y$ is…2020 · 6 Sep · Shift 2 · Q28 · Numerical
A vector $a = α i + 2 j + β k (α, β \in R)$ lies in the plane of the vectors, $b = i + j$ and $c = i - j + 4 k$ …2020 · 7 Jan · Shift 1 · Q26 · MCQ
Let $a$ , $b$ and $c$ be three unit vectors such that $a + b + c = 0$ . If $λ = a . b + b . c + c . a$ …2020 · 7 Jan · Shift 2 · Q35 · MCQ
Let $a = i - 2 j + k$ and $b = i - j + k$ be two vectors. If $c$ is a vector such that $b \times c = b \times a$ …2020 · 8 Jan · Shift 2 · Q36 · MCQ
Let $a$ , $b$ and $c$ be three vectors such that $a = 3$ , $b = 5, b . c = 10$ and…2020 · 9 Jan · Shift 2 · Q29 · Numerical
Let $a \to = 3 i \land + 2 j \land + x k \land$ and $b \to = i \land - j \land + k \land$ , for some real…2019 · 8 Apr · Shift 2 · Q25 · MCQ
Let $α = 3 i + j$ and $β = 2 i - j + 3 k$ . If $β = β_{1} - β_{2}$ , where $β_{1}$ …2019 · 9 Apr · Shift 1 · Q28 · MCQ
If a unit vector $a$ makes angles $π$ /3 with $i$ , $π$ / 4 with $j$ and $θ \in$ (0, $π$ ) with $k$ , then a value of $θ$ is :-2019 · 9 Apr · Shift 2 · Q31 · MCQ
Let $a = i + j + 2 k, b = b_{1} i + b_{2} j + 2 k$ , $c = 5 i + j + 2 k$ be three vectors…2019 · 9 Jan · Shift 2 · Q44 · MCQ
Let A (3, 0, –1), B(2, 10, 6) and C(1, 2, 1) be the vertices of a triangle and M be the midpoint of AC. If G divides BM in the ratio, 2 : 1, then cos ( $∠$ GOA) (O being the origin) is equal to :2019 · 10 Apr · Shift 1 · Q37 · MCQ
The distance of the point having position vector $- i + 2 j + 6 k$ from the straight line passing through the point (2, 3, – 4) and parallel to the vector, $6 i + 3 j - 4 k$ is :2019 · 10 Apr · Shift 2 · Q30 · MCQ
Let $a = 2 i + λ_{1} j + 3 k, b = 4 i + (3 - λ_{2}) j + 6 k,$ and $c = 3 i + 6 j + (λ_{3} - 1) k$ …2019 · 10 Jan · Shift 1 · Q27 · MCQ
If $α$ = $(λ - 2) a + b$ and $β = (4 λ - 2) a + 3 b$ be two given vectors $a$ …2019 · 10 Jan · Shift 2 · Q32 · MCQ
Let $a = i + 2 j + 4 k, b = i + λ j + 4 k$ and $c = 2 i + 4 j + (λ^{2} - 1) k$ be…2019 · 11 Jan · Shift 1 · Q30 · MCQ
Let $3 i + j, i + 3 j$ and $β i + (1 - β) j$ respectively be the position vectors of the points A, B and C with respect to the origin O. If the…2019 · 11 Jan · Shift 2 · Q42 · MCQ
Let $a = 3 i + 2 j + 2 k$ and $b = i + 2 j - 2 k$ be two vectors. If a vector perpendicular to both the vectors $a + b$ …2019 · 12 Apr · Shift 1 · Q37 · MCQ
If $a, b,$ and $C$ are unit vectors such that $a + 2 b + 2 c = 0,$ then $a \times c$ …2018 · 15 Apr · Shift 1 · Q38 · MCQ
If the position vectors of the vertices A, B and C of a $Δ$ ABC are respectively $4 i + 7 j + 8 k, 2 i + 3 j + 4 k,$ and $2 i + 5 j + 7 k,$ then the position…2018 · 15 Apr · Shift 2 · Q30 · MCQ
Let $a = i + j + k, c = j - k$ and a vector $b$ be such that $a \times b = c$ and $a . b = 3.$ …2018 · 16 Apr · Shift 1 · Q43 · MCQ
Let $u$ be a vector coplanar with the vectors $a = 2 i + 3 j - k$ and $b = j + k$ . If $u$ is perpendicular to $a$ …2018 · Shift 0 · Q36 · MCQ
The area (in sq. units) of the parallelogram whose diagonals are along the vectors $8 i - 6 j$ and $3 i + 4 j - 12 k,$ is :2017 · 8 Apr · Shift 1 · Q40 · MCQ
If the vector $b = 3 j + 4 k$ is written as the sum of a vector $b_{1},$ paralel to $a = i + j$ and a vector $b_{2},$ perpendicular to $a,$ …2017 · 9 Apr · Shift 1 · Q35 · MCQ
Let $a = 2 i + j - 2 k$ and $b = i + j$ . Let $c$ be a vector such that $c - a = 3$ , $(a \times b) \times c = 3$ …2017 · Shift 0 · Q35 · MCQ
In a triangle ABC, right angled at the vertex A, if the position vectors of A, B and C are respectively 3 $i$ + $j - k$ , $- i$ + 3 $j$ + p $k$ and 5 $i$ + q $j -$ 4 $k$ …2016 · 9 Apr · Shift 1 · Q34 · MCQ
Let ABC be a triangle whose circumcentre is at P. If the position vectors of A, B, C and P are $a, b, c$ and $\frac{a + b + c}{4}$ …2016 · 10 Apr · Shift 1 · Q32 · MCQ
If the vectors $A B = 3 i + 4 k$ and $A C = 5 i - 2 j + 4 k$ are the sides of a triangle $A B C,$ then the length of the median through $A$ is :2013 · Shift 0 · Q46 · MCQ
Let $a$ and $b$ be two unit vectors. If the vectors $c = a + 2 b$ and $d = 5 a - 4 b$ are perpendicular to each other, then the angle…2012 · Shift 0 · Q28 · MCQ
Let $A B C D$ be a parallelogram such that $A B = q, A D = p$ and $∠ B A D$ be an acute angle. If $r$ is the vector that coincide with the altitude…2012 · Shift 0 · Q44 · MCQ
The vectors $a$ and $b$ are not perpendicular and $c$ and $d$ are two vectors satisfying $b \times c = b \times d$ …2011 · Shift 0 · Q35 · MCQ
Let $a$ , $b$ , $c$ be three non-zero vectors which are pairwise non-collinear. If $a + 3 b$ is collinear with $c$ and $b + 2 c$ …2011 · Shift 0 · Q52 · MCQ
If the vectors $a = i - j + 2 k, b = 2 i + 4 j + k$ and $c = λ i + j + μ k$ are…2010 · Shift 0 · Q49 · MCQ
The non-zero vectors are $a, b$ and $c$ are related by $a = 8 b$ and $c = - 7 b .$ Then the angle between $a$ …2008 · Shift 0 · Q43 · MCQ
If $u$ and $v$ are unit vectors and $θ$ is the acute angle between them, then $2 u \times 3 v$ is a unit vector for :2007 · Shift 0 · Q54 · MCQ
The values of a, for which the points $A, B, C$ with position vectors $2 i - j + k, i - 3 j - 5 k$ and $a i - 3 j + k$ respectively are the vertices of a…2006 · Shift 0 · Q65 · MCQ
For any vector $a$ , the value of $(a \times i)^{2} + (a \times j)^{2} + (a \times k)^{2}$ is equal…2005 · Shift 0 · Q112 · MCQ
If $C$ is the mid point of $A B$ and $P$ is any point outside $A B,$ then :2005 · Shift 0 · Q90 · MCQ
Let $a, b$ and $c$ be distinct non-negative numbers. If the vectors $a i + a j + c k, i + k$ and $c i + c j + b k$ lie in a plane, then $c$ is :2005 · Shift 0 · Q91 · MCQ
Let $a, b$ and $c$ be non-zero vectors such that $(a \times b) \times c = \frac{1}{3} b c a .$ …2004 · Shift 0 · Q113 · MCQ
Let $a, b$ and $c$ be three non-zero vectors such that no two of these are collinear. If the vector $a + 2 b$ is collinear with $c$ and $b + 3 c$ …2004 · Shift 0 · Q115 · MCQ
A particle acted on by constant forces $4 i + j - 3 k$ and $3 i + j - k$ is displaced from the point $i + 2 j + 3 k$ to the point $5 i + 4 j + k .$ …2004 · Shift 0 · Q91 · MCQ
Let $u, v, w$ be such that $u = 1, v 2, w 3.$ If the projection $v$ …2004 · Shift 0 · Q92 · MCQ
If $a \times b = b \times c = c \times a$ then $a + b + c =$ 2003 · Shift 0 · Q89 · MCQ
Let $u = i + j, v = i - j$ and $w = i + 2 j + 3 k .$ If $n$ is a unit vector such that $u . n = 0$ …2003 · Shift 0 · Q90 · MCQ
The vectors $A B = 3 i + 4 k & A C = 5 i - 2 j + 4 k$ are the sides of triangle $A B C .$ The length of the median through $A$ is :2003 · Shift 0 · Q91 · MCQ
$a, b, c$ are $3$ vectors, such that $a + b + c = 0$ , $a = 1 b = 2, c = 3,$ …2003 · Shift 0 · Q94 · MCQ
A tetrahedron has vertices at $O (0, 0, 0), A (1, 2, 1) B (2, 1, 3)$ and $C (- 1, 1, 2) .$ Then the angle between the faces $O A B$ and $A B C$ will be :2003 · Shift 0 · Q95 · MCQ
If $a b c a^{2} b^{2} c^{2} 1 + a^{3} 1 + b^{3} 1 + c^{3} = 0$ and vectors $(1, a, a^{2}), (1, b, b^{2})$ …2003 · Shift 0 · Q96 · MCQ
Consider points $A, B, C$ and $D$ with position vectors $7 i - 4 j + 7 k, i - 6 j + 10 k, - i - 3 j + 4 k$ and $5 i - j + 5 k$ …2003 · Shift 0 · Q97 · MCQ
If $a = 4, b = 2$ and the angle between $a$ and $b$ is $π /6$ then $(a \times b)^{2}$ …2002 · Shift 0 · Q89 · MCQ
If the vectors $a, b$ and $c$ from the sides $B C, C A$ and $A B$ respectively of a triangle $A B C$ , then :2002 · Shift 0 · Q90 · MCQ
If $a = 5, b = 4, c = 3$ thus what will be the value of $a . b + b . c + c . a,$ …2002 · Shift 0 · Q91 · MCQ
$a = 3 i - 5 j$ and $b = 6 i + 3 j$ are two vectors and $c$ is a vector such that $c = a \times b$ then $a : b : c$ …2002 · Shift 0 · Q92 · MCQ
If the vectors $c, a = x i + y j + z k$ and $b = j$ are such that $a, c$ and $b$ form a right handed system then $c$ …2002 · Shift 0 · Q93 · MCQ