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Vector Algebra question

2023 · 30 Jan · Shift 1 · Q31
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  5. /2023 · 30 Jan · Shift 1 · Q31

Vector Algebra question

2023 · 30 Jan · Shift 1 · Q31

JEE MainMathematicsVector AlgebraMCQ+4 / −1
Let a unit vector OP^\widehat{O P}OP make angles α,β,γ\alpha, \beta, \gammaα,β,γ with the positive directions of the co-ordinate axes OX\mathrm{OX}OX, OY,OZ\mathrm{OY}, \mathrm{OZ}OY,OZ respectively, where β∈(0,π2)\beta \in\left(0, \frac{\pi}{2}\right)β∈(0,2π​). If OP^\widehat{\mathrm{OP}}OP is perpendicular to the plane through points (1,2,3),(2,3,4)(1,2,3),(2,3,4)(1,2,3),(2,3,4) and (1,5,7)(1,5,7)(1,5,7), then which one of the following is true?
  1. A
    α∈(π2,π)\alpha \in\left(\frac{\pi}{2}, \pi\right)α∈(2π​,π) and γ∈(π2,π)\gamma \in\left(\frac{\pi}{2}, \pi\right)γ∈(2π​,π)
  2. B
    α∈(0,π2)\alpha \in\left(0, \frac{\pi}{2}\right)α∈(0,2π​) and γ∈(π2,π)\gamma \in\left(\frac{\pi}{2}, \pi\right)γ∈(2π​,π)
  3. C
    α∈(π2,π)\alpha \in\left(\frac{\pi}{2}, \pi\right)α∈(2π​,π) and γ∈(0,π2)\gamma \in\left(0, \frac{\pi}{2}\right)γ∈(0,2π​)
  4. D
    α∈(0,π2)\alpha \in\left(0, \frac{\pi}{2}\right)α∈(0,2π​) and γ∈(0,π2)\gamma \in\left(0, \frac{\pi}{2}\right)γ∈(0,2π​)
View written solutionFree

Correct answer: A

  1. Find two direction vectors in the plane

The plane passes through points A(1,2,3),B(2,3,4),C(1,5,7).A(1,2,3),\quad B(2,3,4),\quad C(1,5,7).A(1,2,3),B(2,3,4),C(1,5,7).

So, two vectors in the plane are AB→=(2−1,3−2,4−3)=(1,1,1),\overrightarrow{AB}= (2-1,3-2,4-3)=(1,1,1),AB=(2−1,3−2,4−3)=(1,1,1), AC→=(1−1,5−2,7−3)=(0,3,4).\overrightarrow{AC}= (1-1,5-2,7-3)=(0,3,4).AC=(1−1,5−2,7−3)=(0,3,4).

  1. Find a normal vector to the plane

A vector perpendicular to the plane is n→=AB→×AC→.\overrightarrow{n}=\overrightarrow{AB}\times\overrightarrow{AC}.n=AB×AC.

Compute:

=\begin{vmatrix} \hat i & \hat j & \hat k\\ 1&1&1\\ 0&3&4 \end{vmatrix}$$ $$=\hat i(1\cdot 4-1\cdot 3)-\hat j(1\cdot 4-1\cdot 0)+\hat k(1\cdot 3-1\cdot 0)$$ $$=\hat i-4\hat j+3\hat k.$$ So one normal vector is $$\overrightarrow{n}=(1,-4,3).$$ Its opposite vector $$(-1,4,-3)$$ is also normal to the plane. 3. **Use the condition on** $\beta$ If $\widehat{OP}$ is a unit vector making angles $\alpha,\beta,\gamma$ with the positive coordinate axes, then its direction cosines are $$\widehat{OP}=(\cos\alpha,\cos\beta,\cos\gamma).$$ Since $\widehat{OP}$ is perpendicular to the plane, it must be along a unit normal vector. Now given $$\beta\in\left(0,\frac{\pi}{2}\right),$$ so $$\cos\beta>0.$$ Thus the $y$-component of $\widehat{OP}$ must be positive. Among the two normal directions: - $(1,-4,3)$ has negative $y$-component, - $(-1,4,-3)$ has positive $y$-component. Hence $\widehat{OP}$ must be in the direction of $$(-1,4,-3).$$ 4. **Determine signs of** $\cos\alpha$ **and** $\cos\gamma$ For the direction $(-1,4,-3)$: - $x$-component is negative $\Rightarrow \cos\alpha<0 \Rightarrow \alpha\in\left(\frac{\pi}{2},\pi\right)$, - $z$-component is negative $\Rightarrow \cos\gamma<0 \Rightarrow \gamma\in\left(\frac{\pi}{2},\pi\right)$. Therefore, $$\alpha\in\left(\frac{\pi}{2},\pi\right),\qquad \gamma\in\left(\frac{\pi}{2},\pi\right).$$ 5. **Match with options** This corresponds to **Option A**.
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