Question

Let
$\stackrel{→}{a} = \hat{i} + \hat{j} + \hat{2k}, \stackrel{→}{b} = \hat{2i} - \hat{3j} + \hat{k}$
and
$\stackrel{→}{c}= \hat{i} - \hat{j} + \hat{k}$
be three given vectors.
Let $\stackrel{→}{v}$ be a vector in the plane of $\stackrel{→}{a}$ and $\stackrel{→}{b}$ whose projection on $\stackrel{→}{c}$ is $\frac{2}{\sqrt3}$.
If $\stackrel{→}{v}.\hat{j}$ = 7 , then $\stackrel{→}{v}.(\hat{i}+\hat{k})$ is equal to :

• A. 6
• B. 7
• C. 8
• D. 9

Answer 1

Answer: (A)

6

Answer 2

The correct answer is (D) : 9
Let
$\stackrel{→}{v} = λ_1\stackrel{→}{a} λ_2\stackrel{→}{b},$
where $λ_1,λ_2 ∈ R$
$= (λ_1 + 2λ_2)\hat{i} + (λ_1 - 3λ_2)\hat{j}+ (2λ_1 + λ_2)\hat{k}$
∵ Projection of $\stackrel{→}{v}$ on $\stackrel{→}{c}$ is $\frac{2}{\sqrt3}$
∴$$\frac{ λ_1 + 2λ_2 - λ_1 + 3λ_2 + 2λ_1 + λ_2}{\sqrt3} = \frac{2}{\sqrt3}
$∴ λ_1 + 3λ_2 = 1 ......... (i)$
and
$\stackrel{→}{v}.\hat{j} = 7 ⇒ λ_1 - 3λ_2 = 7 ......... (ii)$
from equation (i) and (ii)
$λ_1 = 4, λ_2 = -1$
$∴ \stackrel{→}{v} = 2\hat{i} + 7\hat{j} + 7\hat{k}$
$∴ \stackrel{→}{v}.( \hat{i} + \hat{k} ) = 2 + 7$
$= 9$