Question

Let $\vec{ a }=4 \hat{ i }+3 \hat{ j }$ and $\vec{ b }=3 \hat{ i }-4 \hat{ j }+5 \hat{ k }$ and $\vec{ c }$ is a vector such that $\vec{ c } \cdot(\vec{ a } \times \vec{ b })+25=0, \vec{ c } \cdot(\hat{ i }+\hat{ j }+\hat{ k })=4$ and projection of $\vec{ c }$ on $\vec{ a }$ is 1 , then the projection of $\vec{ c }$ on $\vec{ b }$ equals :

• A. $\frac{5}{\sqrt{2}}$
• B. $\frac{1}{5}$
• C. $\frac{1}{\sqrt{2}}$
• D. $\frac{3}{\sqrt{2}}$

Answer 1

Answer: (A)

$\vec{ a } \times \vec{ b }=15 \hat{ i }-20 \hat{ j }-25 \hat{ k }$
Let $ \vec{c}=x \hat{i}+y \hat{j}+z \hat{k}$
$\Rightarrow 15 x-20 y-25 z+25=0$
$\Rightarrow 3 x-4 y-5 z=-5$
Also $x+y+z=4$
and $\frac{\vec{ c } \cdot \vec{ a }}{|\vec{ a }|}=1 \Rightarrow 4 x+3 y=5$
$\Rightarrow \vec{c}=2 \hat{i}-\hat{j}+3 \hat{k}$
Projection of $\vec{ c }$
or $\vec{ b }=\frac{25}{5 \sqrt{2}}=\frac{5}{\sqrt{2}}$