Find the projection of the vector $\hat i + 3\hat j + 7\hat k$ on the vector $7\hat i - \hat j + 8\hat k$
447,Ex.10.3,Q.No 4]

Let $\vec a = \hat i + 3\hat j + 7\hat k$ and $\vec b = 7\hat i - \hat j + 8\hat k$ , $|\vec b| = \sqrt {{7^2} + {{( - 1)}^2} + {8^2}} = \sqrt {49 + 1 + 64} = \sqrt {114}$

and $\vec a\vec b = \left( {\hat i + 3\hat j + 7\hat k} \right) \cdot \left( {7\hat i - \hat j + 8\hat k} \right) = 7 - 3 + 56 = 60$

$\therefore$ Projection of $\vec a$ on $\vec b$ $= \cfrac{{\vec a \cdot \vec b}}{{|\vec b|}} = \cfrac{{60}}{{\sqrt {114} }}$.