If the direction ratios of two vectors are connected by the relations $p + q + r = 0$ and $p^{2} + q^{2} - r^{2} = 0$ , find the acute angle between them in degrees

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Solution

$\begin{matrix} S i n c e, p + q + r = 0 a n d, p^{2} + q^{2} - r^{2} = 0 W e k n o w t h a t p^{2} + q^{2} + r^{2} = 1 2 r^{2} = 1 r = \pm \frac{1}{\sqrt{2}} A l s o, p + q = - r = \pm \frac{1}{\sqrt{2}} {(- p + q)}^{2} = r^{2} p^{2} + q^{2} + 2 p q = r^{2} p q = 0 I f p = 0 2 q^{2} = 1 q = \pm \frac{1}{\sqrt{2}} N o w, i f θ i s t h e a n g l e b e t w e e n l i n e s t h e n cos θ = β_{1} β_{2} + q_{1} q_{2} + r_{1} r_{2} = 0 o r \frac{1}{2} θ = \frac{π}{2} o r \frac{π}{3} \end{matrix}$

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