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Applications of Dot Product

Let a⃗, b⃗,...

Question

Let $\to a, \to b, \to c$ be three unit vectors such that $∣ ∣ \to a + \to b + \to c ∣ ∣ = 1$ and $\to a ⊥ \to b$ . If $\to c$ makes angle $α, β$ with $\to a, \to b$ respectively then $cos α + cos β$ is equal to

A

\frac{3}{2}

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B

1

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C

- 1

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D

None of these

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Solution

The correct option is C $- 1$
Consider
$¯ ¯ ¯ a = i$
$¯ ¯ b = j$
$¯ ¯ c = - i$
Now
$¯ ¯ ¯ a + ¯ ¯ b + ¯ ¯ c$
$= i - i + j$
$= j$
Hence $| ¯ ¯ ¯ a + ¯ ¯ b + ¯ ¯ c |$ $= 1$

If $\to a . \to c = | a | | c | cos α$
$i . (- i) = cos (α)$
$cos α = - 1$
Now
$\to b . \to c = | b | | c | cos β$
$j . (- i) = cos (β)$
$cos β = 0$
Hence
$cos α + cos β = - 1$

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