Let a=2i-5j+5k and b=i-j+3k.

Question

Let $\vec{a}=2\hat{i}-5\hat{j}+5\hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+3\hat{k}$. If $\vec{c}$ is a vector such that $2(\vec{a}	imes \vec{c})+3(\vec{b}	imes \vec{c})=\overset{	o }{0}$ and $(\vec{a}-\vec{b})\cdot \vec{c}=-97$, then $|\vec{c}	imes \hat{k}{|}^{2}$ is equal to

Accepted Answer

Given

$\begin{gathered} \vec{a}=2\hat{i}-5\hat{j}+5\hat{k},\, \vec{b}=\hat{i}-\hat{j}+3\hat{k} \ a(\vec{a}	imes \vec{c})+3(\vec{b}	imes \vec{c})=0 \ 2\vec{a}	imes \vec{c}+3\vec{b}	imes \vec{c}=0 \ (2\vec{a}+3\vec{b})	imes \vec{c}=0 \end{gathered}$

This means $2\vec{a}+3\vec{b}$ and $\vec{c}$ are parallel.

$\Rightarrow \vec{c}=\lambda (2\vec{a}+3\vec{b})$

$\vec{c}=\lambda [2(2\hat{i}-5\hat{j}+5\hat{k})+3(\hat{i}-\hat{j}+3\hat{k})]$

$\vec{c}=\lambda (7\hat{i}-13\hat{j}+19\hat{k})$

Given

$(\vec{a}-\vec{b})\cdot \vec{c}=-97$

$\begin{gathered} \vec{a}-\vec{b}=(2\hat{i}-5\hat{j}+5\hat{k})-(\hat{i}-\hat{j}+3\hat{k}) \ =\hat{i}-4\hat{j}+2\hat{k} \end{gathered}$

Put in equation:

$\begin{gathered} (\hat{i}-4\hat{j}+2\hat{k})\cdot \lambda (7\hat{i}-13\hat{j}+19\hat{k})=-97 \ \lambda (7+52+38)=-97 \ \lambda (97)=-97\Rightarrow \lambda =-1 \ \vec{c}=-1(7\hat{i}-13\hat{j}+19\hat{k}) \ \vec{c}=-7\hat{i}+13\hat{j}-19\hat{k} \ \vec{c}	imes \hat{k}=|\begin{array}{ccc}\hat{i} \quad  \hat{j} \quad  \hat{k} \ -7 \quad  13 \quad  -19 \ 0 \quad  0 \quad  1\end{array}| \ =\hat{i}(13)-\hat{j}(-7)+\hat{k}(0) \ =13\hat{i}+7\hat{j} \ |\vec{c}	imes \hat{k}{|}^{2}={13}^{2}+{7}^{2}=169+49=218 \end{gathered}$

Solutions · 1