JEE Main 2022 · 25 Jun · Shift 1 · Mathematics Q14

← JEE Main 2022 · 25 Jun · Shift 1

MathematicsSingle correct+4 / -1Question 14 of 90

Let $\vec a=a_1\hat i+a_2\hat j+a_3\hat k$ , $a_i>0$ , $i=1,2,3$ be a vector which makes equal angles with the coordinate axes OX, OY and OZ. Also, let the projection of $\vec a$ on the vector $3\hat i+4\hat j$ be 7. Let $\vec b$ be a vector obtained by rotating $\vec a$ with 90°. If $\vec a$ , $\vec b$ and $x$ -axis are coplanar, then projection of a vector $\vec b$ on $3\hat i+4\hat j$ is equal to :

\sqrt7

\sqrt2

Answer and worked solution

Official answer: B

Find $\vec a$ . Equal angles and $a_i>0$ : $\vec a=c(1,1,1)$ ; projection $\dfrac{7c}5=7$ gives $c=5$ , $|\vec a|=5\sqrt3$ .

Find $\vec b$ . In the plane of $\vec a$ and $\hat i$ , perpendicular to $\vec a$ : $\hat i-\dfrac13(1,1,1)\propto(2,-1,-1)$ . With $|\vec b|=5\sqrt3$ : $\vec b=\pm\dfrac5{\sqrt2}(2,-1,-1)$ .

Projection on $3\hat i+4\hat j$ . $\dfrac5{\sqrt2}\cdot\dfrac{6-4}5=\sqrt2$ (taking the positive sense).

(B)

Topics:Dot product Vector operations

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JEE Main 2022 · 25 Jun · Shift 1, Mathematics question 14