Let a - b - c - be three unit vectors- such that a - b - c -1 and a - is perpendicular to b - If c - makes angles - and - witha - and b - respectively- then cos-cos-a -32b 32c 1d -1

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Byju's Answer

Standard XII

Mathematics

Dot Product of Two Vectors

Let a →, b →,...

Question

Let $\vec{a}, \vec{b}, \vec{c}$ be three unit vectors, such that $|\vec{a} + \vec{b} + \vec{c}|$ =1 and $\vec{a}$ is perpendicular to $\vec{b}$ . If $\vec{c}$ makes angles α and β with $\vec{a} and \vec{b}$ respectively, then cos α + cos β =
(a) $- \frac{3}{2}$

(b) $\frac{3}{2}$

(c) 1

(d) −1

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Solution

(d) −1

$Given that \vec{a}, \vec{b} and \vec{c} are unit vectors. So, |\vec{a}| =1, |\vec{b}| =1and |\vec{c}| =1. Since \vec{a} and \vec{b} are mutually perpendicular, \vec{a} . \vec{b} = 0 Now, |\vec{a} + \vec{b} + \vec{c}| = 1 \Rightarrow {|\vec{a} + \vec{b} + \vec{c}|}^{2} = 1 \Rightarrow {|\vec{a}|}^{2} + {|\vec{b}|}^{2} + {|\vec{c}|}^{2} + 2 \vec{a} . \vec{b} + 2 \vec{b} . \vec{c} + 2 \vec{c} . \vec{a} = 1 \Rightarrow 1 + 1 + 1 + 2 (0) + 2 |\vec{a}| |\vec{b}| \cos β + 2 |\vec{c}| |\vec{a}| \cos α = 1 \Rightarrow 3 + 2 (\cos α + \cos β) = 1 \Rightarrow 2 (\cos α + \cos β) = - 2 \Rightarrow \cos α + \cos β = - 1$

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Let a →, b →, c → be three unit vectors, such that a →+b →+c → =1 and a →is perpendicular to b →. If c →makes angles α and β with a →and b →respectively, then cos α + cos β = (a) -32 (b) 32 (c) 1 (d) −1