First slide

Dot product or scalar product of vectors

Question

If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors such that $\vec{a} \cdot \vec{b} = 0 = \vec{a} \cdot \vec{c}$ and the angle between $\vec{b} and \vec{c} is π / 3$ then the value of $|\vec{a} \times \vec{b} - \vec{a} \times \vec{c}|$ is

Moderate

A

½
B

1
C

2
D

none of these

Solution

$\begin{array}{l} ∣ \vec{a} \times \vec{b} - \vec{a} \times {\vec{c}|}^{2} \\ = | \vec{a} \times (\vec{b} - \vec{c}) |^{2} \\ = | \vec{a} |^{2} | \vec{b} - \vec{c} |^{2} - (\vec{a} \cdot (\vec{b} - \vec{c}))^{2} = | \vec{b} - \vec{c} |^{2} \\ = | \vec{b} |^{2} + | \vec{c} |^{2} - 2 | \vec{b} | | \vec{c} | \cos \frac{π}{3} = 1 \end{array}$

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