Let a be a non-zero vector parallel to the line of intersection of the two planes describe…

Question

Let $\vec{a}$ be a non-zero vector parallel to the line of intersection of the two planes described by $\hat{i}+\hat{j},\hat{i}+\hat{k}$ and $\hat{i}-\hat{j},\hat{j}-\hat{k}$. If $	heta $ is the angle between the vector $\vec{a}$ and the vector $\vec{b}=2\hat{i}-2\hat{j}+\hat{k}$ and $\vec{a}\cdot \vec{b}=6$, then the ordered pair $(	heta ,|\vec{a}	imes \vec{b}|)$ is equal to :

Accepted Answer

We have, $\vec{a}$ is non-zero vector parallel to the line of intersection of the two planes described by $\hat{i}+\hat{j},\hat{i}+\hat{k}$ and $\hat{i}-\hat{j},\hat{j}-\hat{k}$.

Let ${n}_{1}$ and ${n}_{2}$ are the normal vector to the plane $\hat{i}+\hat{j},\hat{i}+\hat{k}$ and $\hat{i}-\hat{j},\hat{j}-\hat{k}$, respectively.

$\begin{gathered} {n}_{1}=|\begin{array}{ccc}\hat{i} \quad  \hat{j} \quad  \hat{k} \ 1 \quad  1 \quad  0 \ 1 \quad  0 \quad  1\end{array}|=\hat{i}-\hat{j}-\hat{k} \ {n}_{2}=|\begin{array}{ccc}\hat{i} \quad  \hat{j} \quad  \hat{k} \ 1 \quad  -1 \quad  0 \ 1 \quad  0 \quad  -1\end{array}|=\hat{i}+\hat{j}+\hat{k} \ \vec{a}=\lambda |{n}_{1}	imes {n}_{2}| \end{gathered}$

[ $\because a$ is parallel to line of intersection of both planes]

$\begin{gathered} =\lambda |\begin{array}{ccc}\hat{i} \quad  \hat{j} \quad  \hat{k} \ 1 \quad  -1 \quad  -1 \ 1 \quad  1 \quad  1\end{array}| \ =\lambda (-2\hat{j}+2\hat{k}) \end{gathered}$

$\vec{a}\cdot \vec{b}=6[Given]$

$\lambda (0+4+2)=6$

$\lambda =1$

$\begin{gathered} 	herefore \vec{a} =-2\hat{j}+2\hat{k} \ \ \cos 	heta =\frac{\vec{a}\cdot \vec{b}}{|\vec{a}||\vec{b}|}=\frac{6}{\sqrt{4+4}\sqrt{4+4+1}} \ \ =\frac{1}{\sqrt{2}} \end{gathered}$

$	herefore 	heta =\frac{\pi }{4}$

$\begin{gathered} \vec{a}	imes \vec{b} \quad  =|\begin{array}{ccc}\hat{i} \quad  \hat{j} \quad  \hat{k} \ 0 \quad  -2 \quad  2 \ 2 \quad  -2 \quad  1\end{array}| \ =2\hat{i}+4\hat{j}+4\hat{k} \ |\vec{a}	imes \vec{b}| \quad  =\sqrt{4+16+16}=6 \ (	heta ,|\vec{a}	imes \vec{b}|) \quad  =(\frac{\pi }{4},6) \end{gathered}$

Solutions · 1