First slide

Cross product of vectors or vector product of vectors

Question

If $\vec{a} and \vec{b}$ are orthogonal unit vectors, then for a vector $\vec{r}$ non-coplanar with $\vec{a} and \vec{b}$ vector $\vec{r} \times \vec{a}$ is equal to

Moderate

A

$[\vec{r} \vec{a} \vec{b}] \vec{b} - (\vec{r} \cdot \vec{b}) (\vec{b} \times \vec{a})$
B

$[\vec{r} \vec{a} \vec{b}] (\vec{a} + \vec{b})$
C

$[\vec{r} \vec{a} \vec{b}] \vec{a} + (\vec{r} \cdot \vec{a}) \vec{a} \times \vec{b}$
D

none of these

Solution

$\begin{array}{l} \vec{r} \times \vec{a} = λ \vec{a} + μ \vec{b} + γ \vec{a} \times \vec{b} \\ ∴ [\vec{r} \vec{a} \vec{a}] = λ \vec{a} \cdot \vec{a} + μ \vec{b} \cdot \vec{a} + γ [\vec{a} \vec{b} \vec{a}] \\ 0 = λ | \vec{a} |^{2} + 0 + 0 \\ λ = 0 \end{array}$
$\begin{array}{l} also [\vec{r} \vec{a} \vec{b}] = λ \vec{a} \cdot \vec{b} + μ \vec{b} \cdot \vec{b} + γ [\vec{a} \vec{b} \vec{b}] = μ \\ also (\vec{r} \times \vec{a}) \times \vec{b} = γ (\vec{a} \times \vec{b}) \times \vec{b} \end{array}$
$\begin{matrix} or (\vec{r} \cdot \vec{b}) \vec{a} - (\vec{a} \cdot \vec{b}) \vec{r} = γ {(\vec{a} \cdot \vec{b}) \vec{b} - (\vec{b} \cdot \vec{b}) \vec{a}} \\ or (\vec{r} \cdot \vec{b}) \vec{a} = - γ \vec{a} or γ = - (\vec{r} \cdot \vec{b}) \end{matrix}$

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