Question

Let $\overset{→}a = \hat{i}- \hat{J}+2\hat{K}$ and $\overset{→}b$ be a vector such that $\overset{→}a×\overset{→}b=2\hat{i}−\hat{k}$ and $\overset{→}a⋅\overset{→}b=3$. Then the projection of $\overset{→}b$ on the vector $\overset{→}a-\overset{→}b$ is :

• A. $\frac2{\sqrt{21}}$
• B. $2\sqrt{\frac3{7}}$
• C. $\frac{2}3\sqrt{\frac7{3}}$
• D. $\frac2{3}$

Answer 1

Answer: (A)

$\frac2{\sqrt{21}}$

Answer 2

$\overset{→}a = \hat{i}- \hat{J}+2\hat{K}$
$\overset{→}a×\overset{→}b=2\hat{i}−\hat{k}$
$\overset{→}a⋅\overset{→}b=3$

$|\overset{→}a×\overset{→}b|^2 +|\overset{→}a.\overset{→}b|^2 = |\overset{→}a|^2.|\overset{→}b|^2$

⇒ 5 + 9 = 6$|\overset{→}b|^2$m

⇒|\overset{→}b|^2$$=\frac7{3}

$|\overset{→}a-\overset{→}b| = \sqrt{|\overset{→}a|^2+|\overset{→}b|^2}-2\overset{→}a.\overset{→}b = \sqrt{\frac7{3}}$

Projection of $\overset{→}b$ on $\overset{→}a-\overset{→}b$ is :$\frac{\overset{→}b.(\overset{→}a-\overset{→}b)}{|\overset{→}a-\overset{→}b|}$

=$\frac{\overset{→}b.\overset{→}a-|\overset{→}b|^2}{|\overset{→}a-\overset{→}b|}$

=$\frac{3-\frac7{3}}{\sqrt{\frac7{3}}}$

=$\frac2{\sqrt{21}}$