Let a - and b - be two unit vectors and - be the angle between them- Then- a -b - is a unit vector ifa - 4b - 3c -2 - 3d - 2

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Byju's Answer

Standard XII

Mathematics

Dot Product of Two Vectors

Let a → and b...

Question

Let $\vec{a} and \vec{b}$ be two unit vectors and α be the angle between them. Then, $\vec{a} + \vec{b}$ is a unit vector if
(a) $α = \frac{π}{4}$

(b) $α = \frac{π}{3}$

(c) $α = \frac{2 π}{3}$

(d) $α = \frac{π}{2}$

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Solution

(c) $α = \frac{2 π}{3}$

$\vec{a} and \vec{b} are unit vectors . \Rightarrow |\vec{a}| = |\vec{b}| = 1 . . . (1) Now, \vec{a} . \vec{b} = |\vec{a}| |\vec{b}| \cos α \Rightarrow \vec{a} . \vec{b} = \cos α . . . (2) [Using (1)] Given that |\vec{a} + \vec{b}| = 1 Squaring both sides, we get {|\vec{a} + \vec{b}|}^{2} = 1 \Rightarrow {|\vec{a}|}^{2} + {|\vec{b}|}^{2} + 2 \vec{a} . \vec{b} = 1 \Rightarrow 1 + 1 + 2 \cos α = 1 [Using (1) and (2)] \Rightarrow 2 + 2 \cos α = 1 \Rightarrow 2 \cos α = - 1 \Rightarrow 2 \cos α = - 1 \Rightarrow \cos α = \frac{- 1}{2} \Rightarrow α = \frac{2 π}{3}$

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Let a →and b →be two unit vectors and α be the angle between them. Then, a →+b →is a unit vector if (a) α=π4 (b) α=π3 (c) α=2π3 (d) α=π2

Let $\vec{a} and \vec{b}$ be two unit vectors and α be the angle between them. Then, $\vec{a} + \vec{b}$ is a unit vector if
(a) $α = \frac{π}{4}$

(b) $α = \frac{π}{3}$

(c) $α = \frac{2 π}{3}$

(d) $α = \frac{π}{2}$