If θ is the angle between the lines $AB$ and $AC$ where $A,B$ and $C$ are the three points with coordinates $(1,2, - 1),(2,0,3),(3, - 1,2)$ respectively, then $\sqrt{462}\cos heta$ is equal to

Question

If θ is the angle between the lines $AB$ and $AC$ where $A,B$ and $C$ are the three points with coordinates $(1,2, - 1),(2,0,3),(3, - 1,2)$ respectively, then $\sqrt{462}\cos	heta$ is equal to

Dr. Abubacker · Accepted Answer

(a)
$\overrightarrow{\mathbf{AB}} = (2 - 1)\widehat{\dot{\mathbf{i}}} + (0 - 2)\widehat{\dot{\mathbf{j}}} + (3 + 1)\widehat{\mathbf{k}}$
$= \widehat{\dot{\mathbf{i}}} - 2\widehat{\dot{\mathbf{j}}} + 4\widehat{\mathbf{k}}$
and
$\ \overrightarrow{\mathbf{AC}} = (3 - 1)\widehat{\dot{\mathbf{i}}} + ( - 1 - 2)\widehat{\dot{\mathbf{j}}} + (2 + 1)\widehat{\mathbf{k}}$
$= 2\widehat{\dot{\mathbf{i}}} - 3\widehat{\dot{\mathbf{j}}} + 3\widehat{\mathbf{k}}$
$\cos	heta = \frac{\left( \widehat{\dot{\mathbf{i}}} - 2\widehat{\dot{\mathbf{j}}} + 4\widehat{\mathbf{k}} ight) \bullet \left( 2\widehat{\dot{\mathbf{i}}} - 3\widehat{\dot{\mathbf{j}}} + 3\widehat{\mathbf{k}} ight)}{\sqrt{1 + 4 + 16}\sqrt{4 + 9 + 9}}$
$= \frac{2 + 6 + 12}{\sqrt{21}\sqrt{22}} = \frac{20}{\sqrt{462}}$
$\Longrightarrow \ \sqrt{462}\cos	heta = 20$

Q. If θ is the angle between the lines $AB$ and $AC$ where $A,B$ and $C$ are the three points with coordinates $(1,2, - 1),(2,0,3),(3, - 1,2)$ respectively, then $\sqrt{462}\cos\theta$ is equal to

Explanation

Reviewed by Dr. Abubacker

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