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{\bf Question}

The vectors OP and OQ are given respectively by the quantities $
2{\bf i} + 2{\bf j} - 5{\bf k}$ and $ 4{\bf i} - 3{\bf j} + 2{\bf
k}$.  Find PQ, and determine its length and direction cosines.

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{\bf Answer}

$\vec{OP} = 2{\bf i} + 2{\bf j} - 5{\bf k}$ and $\vec{OQ} = 4{\bf
i} - 3{\bf j} + 2{\bf k}$

$\vec{PQ} = \vec{OQ} - \vec{OP} = (4{\bf i} - 3{\bf j} + 2{\bf k})
-  (2{\bf i} + 2{\bf j} -5{\bf k}) = 2{\bf i} - 5{\bf j} + 7{\bf
k}$

length of $\vec{PQ} = |\vec{PQ}| = \sqrt{2^2 + 5^2 + 7^2} =
\sqrt{78}$

$$\begin{array}{lccc} & \cos \alpha = \frac{x}{r} & \cos \beta =
\frac{y}{r} & \cos \gamma = \frac{z}{r} \\ \\  \Rightarrow & \cos
\alpha = \frac{2}{\sqrt{78}} & \cos \beta = \frac{-5}{\sqrt{78}} &
\cos \gamma = \frac {7}{\sqrt{78}} \end {array}$$


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