Find the unit vector in the direction of vect | vedclass

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Question

(A) The given points are $P(1, 2, 3)$ and $Q(4, 5, 6).$The vector $\overrightarrow{PQ}$ is given by $\overrightarrow{PQ} = (4-1)\hat{i} + (5-2)\hat{j}

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$\frac{1}{\sqrt{3}}\hat{i} + \frac{1}{\sqrt{3}}\hat{j} + \frac{1}{\sqrt{3}}\hat{k}$

Vedclass · Answer

(A) The given points are $P(1, 2, 3)$ and $Q(4, 5, 6).$The vector $\overrightarrow{PQ}$ is given by $\overrightarrow{PQ} = (4-1)\hat{i} + (5-2)\hat{j} + (6-3)\hat{k} = 3\hat{i} + 3\hat{j} + 3\hat{k}.$The magnitude of $\overrightarrow{PQ}$ is $|\overrightarrow{PQ}| = \sqrt{3^2 + 3^2 + 3^2} = \sqrt{9 + 9 + 9} = \sqrt{27} = 3\sqrt{3}.$The unit vector in the direction of $\overrightarrow{PQ}$ is $\hat{u} = \frac{\overrightarrow{PQ}}{|\overrightarrow{PQ}|}.$$\hat{u} = \frac{3\hat{i} + 3\hat{j} + 3\hat{k}}{3\sqrt{3}} = \frac{3}{3\sqrt{3}}\hat{i} + \frac{3}{3\sqrt{3}}\hat{j} + \frac{3}{3\sqrt{3}}\hat{k} = \frac{1}{\sqrt{3}}\hat{i} + \frac{1}{\sqrt{3}}\hat{j} + \frac{1}{\sqrt{3}}\hat{k}.$

Find the unit vector in the direction of vector $\overrightarrow{PQ},$ where $P$ and $Q$ are the points $(1, 2, 3)$ and $(4, 5, 6),$ respectively.

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