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  4. The summation of two unit vectors is a third unit vector, then the modulus of the difference of these unit vectors is

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Q. The summation of two unit vectors is a third unit vector, then the modulus of the difference of these unit vectors is

Vector Algebra

Solution:

Since, $\vec{a} + \vec{b} = \vec{c}$
$\Rightarrow \left( \vec{a} + \vec{b}\right)^{2} = \vec{c}^{2}$
$\Rightarrow \left|\vec{a}\right|^{2} + \left|\vec{b}\right|^{2} + 2 \left|\vec{a}\right| \left|\vec{b}\right|cos\theta = \left|\vec{c}\right|^{2}$
$\Rightarrow 2\left(1+cos\theta\right) = 1$
$\Rightarrow cos\theta = -\frac{1}{2}$
Now, $\left|\vec{a} - \vec{b}\right|^{2} = \left|\vec{a}\right|^{2} + \left|\vec{b}\right|^{2} - 2\left|\vec{a}\right| \left|\vec{b}\right|cos\theta $
$= 1 + 1-2\cdot\left(\frac{-1}{2}\right)$
$\Rightarrow \left|\vec{a} - \vec{b}\right| = \sqrt{3}$