Find the projection of the vector $i - 2 j + k$ on the vector $4 i - 4 j + 7 k$

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Solution

Let $\to a = i - 2 j + k$ and $\to b = 4 i - 4 j + 7 k$
Now $\to a \cdot \to b = (^i - 2^j +^k) (4^i - 4^j + 7^k)$
$\Rightarrow \to a . \to b = 4 + 8 + 7 = 19$

Also, $| \to a | = \sqrt{1 + 4 + 1} = \sqrt{6}, | \to b | = \sqrt{16 + 16 + 49} = \sqrt{81} = 9$
Therefore projection of $\to a$ on $\to b$ is $= \frac{| \to a \cdot \to b |}{| \to b |} = \frac{19}{9}$

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