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# Find the projection of $\overrightarrow a = 2\hat i - \hat j + \hat k\: on \: \overrightarrow b = \hat i - 2\hat j + \hat k.$

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Toolbox:
• Projection of $\overrightarrow {a}$ on $\overrightarrow b$ is $\large\frac{\overrightarrow a .\overrightarrow b}{\mid \overrightarrow a\mid}$
Step 1:
$\overrightarrow a=2\hat i-\hat j+\hat k$
$\overrightarrow b=\hat i-2\hat j+\hat k$
$\overrightarrow a.\overrightarrow b=(2\hat i-\hat j+\hat k).(\hat i-2\hat j+\hat k)$
$\qquad\;=2+2+1$
$\qquad\;=5$
$\mid\overrightarrow a\mid=\sqrt{(2)^2+(-1)^2+(1)^2}$
$\qquad=\sqrt{4+1+1}$
$\qquad=\sqrt{6}$
Step 2:
Projection of $\overrightarrow {a}$ on $\overrightarrow b$ is $\large\frac{\overrightarrow a .\overrightarrow b}{\mid \overrightarrow a\mid}$
$\Rightarrow \large\frac{5}{\sqrt 6}$
$\Rightarrow \large\frac{5\sqrt 6}{ 6}$
answered Nov 7, 2013