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Vector Algebra

39 questions · Mathematics · JEE Advanced

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PYQ Library
JEE Advanced
Mathematics
Vector Algebra

Vector Algebra

39 questions · Mathematics · JEE Advanced

For any two points $M$ and $N$ in the $X Y$ -plane, let $M N$ denote the vector from $M$ to $N$ , and $0$ denote the zero vector. Let $R$ and $R$ be three distinct points in the $X Y$ -plane. Let $S$ be a point inside…2025 · Shift 1 · Q25 · Numerical
Let $w = \hat{i} + \hat{j} - 2 \hat{k}$ , and $u$ and $v$ be two vectors, such that $u \times v = w$ and $v \times w = u$ . Let $α, β, γ$ , and $t$ be real numbers such that $u = α \hat{i} + β \hat{j} + γ \hat{k}, - t α + β + γ = 0, α - tβ + γ = 0, α + β - t γ = 0.$ … Includes table2025 · Shift 1 · Q32 · MCQ
Consider the vectors $x = \overset{}{^} + 2 \overset{}{^} + 3 \hat{k}, y = 2 \overset{}{^} + 3 \overset{}{^} + \hat{k}, and z = 3 \overset{}{^} + \overset{}{^} + 2 \hat{k}$ For two distinct positive real numbers $α$ …2025 · Shift 2 · Q28 · Numerical
Let $O P = \frac{α - 1}{α} \hat{i} + \hat{j} + \hat{k}, O Q = \hat{i} + \frac{β - 1}{β} \hat{j} + \hat{k}$ and $O R = \hat{i} + \hat{j} + \frac{1}{2} \hat{k}$ be three vectors, where $α, β \in R - {0}$ …2024 · Shift 1 · Q29 · Numerical
Let $p = 2 \hat{i} + \hat{j} + 3 \hat{k}$ and $q = \hat{i} - \hat{j} + \hat{k}$ . If for some real numbers $γ$ , and $γ$ , we have $15 \hat{i} + 10 \hat{j} + 6 \hat{k} = α (2 p + q) + β (p - 2 q) + γ (p \times q),$ …2024 · Shift 2 · Q28 · Numerical
Let $P$ be the plane $3 x + 2 y + 3 z = 16$ and let $S = {α \hat{i} + β \hat{j} + γ \hat{k} : α^{2} + β^{2} + γ^{2} = 1$ and the distance of $(α, β, γ)$ from the plane $P$ is…2023 · Shift 1 · Q29 · Numerical
Let the position vectors of the points $P, Q, R$ and $S$ be $a = \hat{i} + 2 \hat{j} - 5 \hat{k}, b = 3 \hat{i} + 6 \hat{j} + 3 \hat{k}$ , $c = \frac{17}{5} \hat{i} + \frac{16}{5} \hat{j} + 7 \hat{k}$ and $\vec{d}=2…2023 · Shift 2 · Q21 · MCQ
Let $\overset{}{^}, \overset{}{^}$ and $\hat{k}$ be the unit vectors along the three positive coordinate axes. Let $a = 3 \overset{}{^} + \overset{}{^} - \hat{k}, b = \overset{}{^} + b_{2} \overset{}{^} + b_{3} \hat{k}, b_{2}, b_{3} \in R, c = c_{1} \overset{}{^} + c_{2} \overset{}{^} + c_{3} \hat{k}, c_{1}, c_{2}, c_{3} \in R$ …2022 · Shift 2 · Q31 · Multiple correct
Let $u$ , $v$ and $w$ be vectors in three-dimensional space, where $u$ and $v$ are unit vectors which are not perpendicular to each other and $\overrightarrow…2021 · Shift 1 · Q38 · Numerical
Let O be the origin and $O A = 2 i + 2 j + k$ and $O B = i - 2 j + 2 k$ and $O C = \frac{1}{2} (O B - λ O A)$ …2021 · Shift 2 · Q24 · Multiple correct
Let a and b be positive real numbers. Suppose $P Q = a i + b j$ and $P S = a i - b j$ are adjacent sides of a parallelogram PQRS. Let u and v be the projection vectors of $w = i + j$ along PQ…2020 · Shift 2 · Q29 · Multiple correct
Let $a = 2 i + j - k$ and $b = i + 2 j + k$ be two vectors. Consider a vector c = $α a$ + $β b$ , $α$ , $β \in$ …2019 · Shift 2 · Q27 · Numerical
Let a and b be two unit vectors such that a . b = 0. For some x, y $\in$ R, let $c = x a + y b + a \times b$ . If | $c$ | = 2 and the vector c is…2018 · Shift 1 · Q30 · Numerical
Let O be the origin and let PQR be an arbitrary triangle. The point S is such that $O P$ . $O Q$ + $O R$ . $O S$ = $O R$ .…2017 · Shift 2 · Q24 · MCQ
Let O be the origin and $O X$ , $O Y$ , $O Z$ be three unit vectors in the directions of the sides $QR$ , $R P$ , $P Q$ respectively, of a…2017 · Shift 2 · Q34 · MCQ
Let $u = u_{1} i + u_{2} j + u_{3} k$ be a unit vector in $R^{3}$ and $w = \frac{1}{6} (i + j + 2 k) .$ Given that there exists a vector $v$ …2016 · Shift 2 · Q21 · Multiple correct
Match the following : Column $I$ (A) In $R^{2},$ If the magnitude of the projection vector of the vector $α i + β j$ on $3 i + j$ and If $α = 2 + 3 β,$ …2015 · Shift 1 · Q23 · MCQ
Let $Δ P QR$ be a triangle. Let $a = QR, b = R P$ and $c = P Q .$ If $a = 12, b = 43, b . c = 24,$ …2015 · Shift 1 · Q35 · Multiple correct
Suppose that $p, q$ and $r$ are three non-coplanar vectors in $R^{3}$ . Let the components of a vector $s$ along $p, q$ and $r$ …2015 · Shift 2 · Q23 · Numerical
Let $x, y$ and $z$ be three vectors each of magnitude $2$ and the angle between each pair of them is $\frac{π}{3}$ . If $a$ is a non-zero vector perpendicular to $x$ …2014 · Shift 1 · Q30 · Multiple correct
Let $a, b$ and $c$ be three non-coplanar unit vectors such that the angle between every pair of them is $\frac{π}{3} .$ If $a \times b + b \times c = p a + q b + r c,$ …2014 · Shift 1 · Q31 · Numerical
Let $PR = 3 \hat{i} + \hat{j} - 2 \hat{k}$ and $SQ = \hat{i} - 3 \hat{j} - 4 \hat{k}$ determine diagonals of a parallelogram $P QR S$ and $PT = \hat{i} + 2 \hat{j} + 3 \hat{k}$ …2013 · Shift 1 · Q40 · MCQ
match List $I$ with List $I I$ and select the correct answer using the code given below the lists: List $I$ (P.) Volume of parallelopiped determined by vectors $a, b$ and…2013 · Shift 2 · Q21 · MCQ
If $a, b$ and $c$ are unit vectors satisfying $a - b^{2} + b - c^{2} + c - a^{2} = 9,$ …2012 · Shift 1 · Q23 · Numerical
If $a$ and $b$ are vectors such that $a + b = 29$ and $a \times (2 i + 3 j + 4 k) = (2 i + 3 j + 4 k) \times b,$ …2012 · Shift 2 · Q24 · MCQ
Let $a = i + j + k, b = i - j + k$ and $c = i - j - k$ be three vectors. A vector $v$ in…2011 · Shift 1 · Q38 · MCQ
The vector (s) which is/are coplanar with vectors $i + j + 2 k$ and $i + 2 j + k,$ and perpendicular to the vector $i + j + k$ is/are2011 · Shift 1 · Q39 · Multiple correct
Match the statements given in Column - $I$ with the values given in Column- $I I .$ Column- $I$ (A) If $a = j + 3 k, b = - j + 3 k$ …2011 · Shift 2 · Q30 · MCQ
Let $a = - i - k, b = - i + j$ and $c = i + 2 j + 3 k$ be three given vectors. If $r$ is a vector such that $r \times b = c \times b$ …2011 · Shift 2 · Q31 · Numerical
If $a$ and $b$ are vectors in space given by $a = \frac{i - 2 j}{5}$ and $b = \frac{2 i + j + 3 k}{14},$ …2010 · Shift 1 · Q42 · Numerical
Let $P, Q, R$ and $S$ be the points on the plane with position vectors $- 2 i - j, 4 i, 3 i + 3 j$ and $- 3 i + 2 j$ respectively. The quadrilateral $P QR S$ must be a2010 · Shift 1 · Q44 · MCQ
Two adjacent sides of a parallelogram $A B C D$ are given by $A B = 2 i + 10 j + 11 k$ and $A D = - i + 2 j + 2 k$ The side $A D$ is rotated by an acute…2010 · Shift 2 · Q38 · MCQ
If $a, b, c$ and $d$ are unit vectors such that $(a \times b) . (c \times d) = 1$ and $a . c = \frac{1}{2}$ …2009 · Shift 1 · Q24 · MCQ
The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors $a, b, c$ such that $a . b = b . c = c . a = \frac{1}{2} .$ …2008 · Shift 1 · Q27 · MCQ
Consider the lines, $L_{1} : \frac{x + 1}{3} = \frac{y + 2}{1} = \frac{z + 1}{2} L_{2} : \frac{x - 2}{1} = \frac{y - 2}{2} = \frac{z - 3}{3}$ The shortest distance between $L_{1}$ and $L_{2}$ is :2008 · Shift 2 · Q25 · MCQ
Consider the lines $L_{1} : \frac{x + 1}{3} = \frac{y + 2}{1} = \frac{z + 1}{2} L_{2} : \frac{x - 2}{1} = \frac{y + 2}{2} = \frac{z - 3}{3}$ The unit vector perpendicular to both $L_{1}$ and $L_{2}$ is :2008 · Shift 2 · Q27 · MCQ
Let two non-collinear unit vectors $a$ and $b$ form an acute angle. A point $P$ moves so that at any time $t$ the position vector $O P$ (where $O$ is the origin) is given by $a cos t + b sin t .$ …2008 · Shift 2 · Q28 · MCQ
The number of distinct real values of $λ$ , for which the vectors $- λ^{2} i + j + k, i - λ^{2} j + k$ and $i + j - λ^{2} k$ are…2007 · Shift 1 · Q29 · MCQ
Let the vector $P Q, QR, R S, S T, T U$ and $U P$ , represent the sides of a regular hexagon. Statement 1 : $P Q \times (R S + S T) e 0$ …2007 · Shift 1 · Q34 · MCQ