Question

Let $\vec{a}=\hat{i}-\hat{j}+2 \hat{k}$ and $\vec{b}$ be a vector such that $\vec{a} \times \vec{b}=2 \hat{i}-\hat{k}$ and $\vec{a} \cdot \vec{b}=3$ Then the projection of $\vec{b}$ on the vector $\vec{a}-\vec{b}$ is :-

• A. $\frac{2}{\sqrt{21}}$
• B. $2 \sqrt{\frac{3}{7}}$
• C. $\frac{2}{3} \sqrt{\frac{7}{3}}$
• D. $\frac{2}{3}$

Answer 1

Answer: (A)

$\frac{2}{\sqrt{21}}$

Answer 2

Projection of b on vector a−b is
=∣a−b∣b⋅(a−b)​
=a2+b2−2a⋅b​a⋅b−∣b∣2​=6+b2−6​3−b2​=b3−b2​
∣a×b∣2=5
a2b2−(a⋅b)2=5
6b2=14⇒b2=37​
∴b3−b2​=37​​3−37​​=2×21​