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# If → a and → B Are Two Unit Vectors Such that → a + → B is Also a Unit Vector, Then Find the Angle Between → a and → B - CBSE (Arts) Class 12 - Mathematics

ConceptProduct of Two Vectors Scalar (Or Dot) Product of Two Vectors

#### Question

If $\vec{a}$ and $\vec{b}$ are two unit vectors such that $\vec{a} + \vec{b}$ is also a unit vector, then find the angle between $\vec{a}$ and $\vec{b}$

#### Solution

Let the angle between $\vec{a}$ and $\vec{b}$ be $\theta$.

It is given that $\left| \vec{a} \right| = \left| \vec{b} \right| = \left| \vec{a} + \vec{b} \right| = 1$

$\left| \vec{a} + \vec{b} \right| = 1$

$\Rightarrow \left| \vec{a} + \vec{b} \right|^2 = 1$

$\Rightarrow \left| \vec{a} \right|^2 + 2\left| \vec{a} \right|\left| \vec{b} \right|\cos\theta + \left| \vec{b} \right|^2 = 1$

$\Rightarrow 1 + 2 \times 1 \times 1 \times \cos\theta + 1 = 1$

$\Rightarrow 2\cos\theta = - 1$

$\Rightarrow \cos\theta = - \frac{1}{2} = \cos\frac{2\pi}{3}$

$\Rightarrow \theta = \frac{2\pi}{3}$

Thus, the angle between $\vec{a}$ and $\vec{b}$  is $\frac{2\pi}{3}$

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Solution If → a and → B Are Two Unit Vectors Such that → a + → B is Also a Unit Vector, Then Find the Angle Between → a and → B Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors.
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