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 coordinates axes OX, OY and OZ. Also, let the projection of $\overrightarrow{\mathrm{a}}$ on the vector $3 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}$ be 7. Let $\vec{b}$ be a vector obtained by rotating $\vec{a}$ with $90^{\circ}$. 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

Vector Algebra — JEE Main · 2022 · 25 Jun 2022 · Shift 1 Question with Solution

Let

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 coordinates axes OX, OY and OZ. Also, let the projection of

a

on the vector

3 \hat{i} + 4 \hat{j}

be 7. Let

b

be a vector obtained by rotating

a

with

9 0^{\circ}

.

If

a, b

and

x

-axis are coplanar, then projection of a vector

b

3 \hat{i} + 4 \hat{j}

is equal to

Answer: (B)

Since

a

makes equal angles with coordinate axes,

a = λ (\frac{1}{3} \hat{i} + \frac{1}{3} \hat{j} + \frac{1}{3} \hat{k}) = \frac{λ}{3} (\hat{i} + \hat{j} + \hat{k})

.

Projection of

a

3 \hat{i} + 4 \hat{j}

is 7:

\frac{a \cdot ( 3 i ^ + 4 j ^ )}{∣3 i ^ + 4 j ^ ∣} = 7

\frac{λ}{3} \cdot \frac{7}{5} = 7 \Rightarrow λ = 53

a = 5 (\hat{i} + \hat{j} + \hat{k})

.

Since

a, b

and

x

-axis are coplanar and

a ⊥ b

, let

b = 5 α (\hat{i} + \hat{j} + \hat{k}) + β \hat{i}

.

From

a \cdot b = 0

75 α + 5 β = 0 \Rightarrow β = - 15 α

b = 5 α (- 2 \hat{i} + \hat{j} + \hat{k})

.

Since

∣ b ∣ = 53

5∣ α ∣ 6 = 53 \Rightarrow α = \pm \frac{1}{2}

Projection of

b

3 \hat{i} + 4 \hat{j}

\frac{b \cdot ( 3 i ^ + 4 j ^ )}{5} = \pm \frac{5}{2} \cdot \frac{- 2}{5} = \pm 2

.

Taking absolute value:

2

JEE Main · 2022 · 25 Jun 2022 · Shift 1 · Single correct option