First slide

Cross product of vectors or vector product of vectors

Question

$If 2 \hat{a} + \vec{b} = \hat{a} \times \vec{b} (where \hat{a} is a unit vector), then | 3 \vec{b} + 4 \hat{a} | =$

Moderate

Solution

We have $2 \hat{a} + \hat{b} = \hat{a} \times \hat{b} - - (1)$
$Taking dot product with \hat{a} \times \hat{b}, we get$
$\begin{aligned} (\hat{a} \times \hat{b}) \cdot (\hat{a} \times \hat{b}) = 0 \\ \Rightarrow \vec{b} = λ \hat{a} \end{aligned}$
$So, from (1), λ = - 2 \Rightarrow \vec{b} = - 2 \hat{a}$
$∴ | 3 \vec{b} + 4 \hat{a} | = | 3 (- 2 \hat{a}) + 4 \hat{a} | = | 2 \hat{a} | = 2$

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