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Shortest Distance between Two Skew Lines

a and b are...

Question

$\to a$ and $\to b$ are units vectors inclined to each other at an angle $α, α \in (0, π)$ and $∣ ∣ ∣ \to a + \to b ∣ ∣ ∣ < 1$ then $α \in$

A

(\frac{π}{3}, \frac{2 π}{3})

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B

(\frac{2 π}{3}, π)

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C

(0, \frac{π}{3})

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D

(\frac{π}{4}, \frac{3 π}{4})

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Solution

The correct option is A $(\frac{π}{3}, \frac{2 π}{3})$
REF.Image.
$\to a + \to b = c o s α | 1 | | 1 |$
$= c o s α$
$∣ ∣ \to a + \to b ∣ ∣ < 1$
$0 ⩽ {∣ ∣ \to a + \to b ∣ ∣}^{2} < 1$
$0 ⩽ (\to a + \to b) (\to a + \to b) < 1$
$0 \leq 1 + 1 + 2 c o s α < 1$
$- 2 \leq 2 c o s α < - 1$
$- 1 \leq c o s α < \frac{- 1}{2}$
$α ϵ (\frac{2 π}{3}, π) .$
A is correct.

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