The direction ratios of two lines are given byp,q,r and 1qr,1rp,1pq. Ifis the angle between them such that 0≤α<π, then the value of cosα+cos2α+cos3α+....+cospα⋅cosqα⋅cosrα=

The direction ratios of two lines are given by $〈p, q, r〉 and 〈\frac{1}{q r}, \frac{1}{r p}, \frac{1}{p q}〉$ . Ifis the angle between them such that $0 \leq α < π$ , then the value of $\cos α + \cos^{2} α + \cos^{3} α + .... + \cos^{p} α \cdot \cos^{q} α \cdot \cos^{r} α =$

A
$p q r$
B
$p + q + r$
C
$p - q - r$
D
None

Solution:

The direction rations of two lines are given as $〈p, q, r〉 and 〈\frac{1}{q r}, \frac{1}{r p}, \frac{1}{p q}〉$

Multiply the second set of direction rations with $p q r$ , then the direction ratios are $〈p, q, r〉$

It implies that both lines are parallel.

Hence $α = 0 °$

Therefore, the required value is $1 + 1 + 1 + .... + (p + q + r times) = p + q + r$

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