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Collinear Vectors

The number of...

Question

The number of integral values of $p,$ such that the angle between vectors $\to a = p^i + 3^j - 7^k$ and $\to b = p^i - p^j + 4^k$ is obtuse, is

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Solution

We know, $cos θ = \frac{\to a \cdot \to b}{| \to a | | \to b |} < 0$
( $∵$ angle between both vectors is obtuse)
$\Rightarrow \frac{p^{2} - 3 p - 28}{\sqrt{p^{2} + 9 + 49} \cdot \sqrt{p^{2} + p^{2} + 16}} < 0 \Rightarrow p^{2} - 3 p - 28 < 0 \Rightarrow (p - 7) (p + 4) < 0 p \in (- 4, 7)$
So, number of integral values of $p$ are $10.$

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(\to a + \to b) \cdot (- 7^i + 2^j + 3^k)

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Collinear Vectors

Standard XII Mathematics

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