First slide

Direction ratios and direction cosines

Question

If $A (0, 4, 1), B (2, 3, - 1), C (4, 5, 0)$ respectively, then the angle between $\vec{A B}, \vec{B C}$ is $\frac{α}{2}$ where $α =$

Moderate

A

$π$
B

$\frac{π}{2}$
C

$\frac{π}{4}$
D

$2 π$

Solution

The direction ratios of the line joining the points $A$ and $B$ is $〈2, - 1, - 2〉$
The direction ratios of the line joining the points $B$ and $C$ is $〈2, 2, 1〉$
If $θ$ is the acute angle between two lines having direction ratios $〈a_{1}, b_{1}, c_{1}〉$ and $〈a_{2}, b_{2}, c_{2}〉$
then $\cos θ = |\frac{a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2}}{\sqrt{{a_{1}}^{2} + {b_{1}}^{2} + {c_{1}}^{2}} \cdot \sqrt{{a_{2}}^{2} + {b_{2}}^{2} + {c_{2}}^{2}}}|$
Given that the angle between the lines $\vec{A B}, \vec{B C}$ is $\frac{α}{2}$
It implies that $\cos (\frac{α}{2}) = (\frac{4 - 2 - 2}{\sqrt{9} \sqrt{9}}) = 0$
Hence, $\frac{α}{2} = \frac{π}{2} \Rightarrow α = π$

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