Let a unit vector u=xi+yj+zk make angles ( )/(2),( )/(3) and (2 )/(3) with the vectors 1√2…

Question

Let a unit vector $\hat{u}=x\hat{i}+y\hat{j}+z\hat{k}$ make angles $\frac{\pi }{2},\frac{\pi }{3}$ and $\frac{2\pi }{3}$ with the vectors $\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{k},\frac{1}{\sqrt{2}}\hat{j}+\frac{1}{\sqrt{2}}\hat{k}$ and $\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{j}$ respectively. If $\vec{v}=\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{j}+\frac{1}{\sqrt{2}}\hat{k}$ then $|\hat{u}-\vec{v}{|}^{2}$ is equal to

Accepted Answer

Unit vector $\hat{u}=x\hat{i}+y\hat{j}+z\hat{k}$

$\begin{gathered} {\overset{	o }{p}}_{1}=\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{k},{\overset{	o }{p}}_{2}=\frac{1}{\sqrt{2}}\hat{j}+\frac{1}{\sqrt{2}}\hat{k} \ {\overset{	o }{p}}_{3}=\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{j} \end{gathered}$

Now angle between $\hat{u}$ and ${\overset{	o }{p}}_{1}=\frac{\pi }{2}$

$\hat{u}\cdot {\overset{	o }{p}}_{1}=0\Rightarrow \frac{x}{\sqrt{2}}+\frac{z}{\sqrt{2}}=0$

$\Rightarrow x+z=0$ ...... (i)

Angle between $\hat{u}$ and ${\overset{	o }{p}}_{2}=\frac{\pi }{3}$

$\hat{u}\cdot {\overset{	o }{p}}_{2}=|\hat{u}|\cdot |{\overset{	o }{p}}_{2}|\cos \frac{\pi }{3}$

$\Rightarrow \frac{y}{\sqrt{2}}+\frac{z}{\sqrt{2}}=\frac{1}{2}\Rightarrow y+z=\frac{1}{\sqrt{2}}$ ...... (ii)

Angle between $\hat{u}$ and ${\overset{	o }{p}}_{3}=\frac{2\pi }{3}$

$\hat{u}\cdot {\overset{	o }{p}}_{3}=|\hat{u}|\cdot |{\overset{	o }{p}}_{3}|\cos \frac{2\pi }{3}$

$\Rightarrow \frac{x}{\sqrt{2}}+\frac{4}{\sqrt{2}}=\frac{-1}{2}\Rightarrow x+y=\frac{-1}{\sqrt{2}}$ ..... (iii)

from equation (i), (ii) and (iii) we get

$x=\frac{-1}{\sqrt{2}}\, y=0\, z=\frac{1}{\sqrt{2}}$

$\begin{gathered} \hat{u}-\overset{	o }{v}=\frac{-1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{k}-\frac{1}{\sqrt{2}}\hat{i}-\frac{1}{\sqrt{2}}\hat{j}-\frac{1}{\sqrt{2}}\hat{k} \ \hat{u}-\overset{	o }{v}=\frac{-2}{\sqrt{2}}\hat{i}-\frac{1}{\sqrt{2}}\hat{j} \ 	herefore |\hat{u}-\overset{	o }{v}{|}^{2}={(\sqrt{\frac{4}{2}+\frac{1}{2}})}^{2}=\frac{5}{2} \end{gathered}$

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