Mathematics

Problem

\mathbf{a} \cdot \mathbf{b}

and

\mathbf{c}

are any three non-zero vectors, then the component of

\mathbf{a} \times (\mathbf{b} \times \mathbf{c})

perpendicular to

\mathbf{b}

is a.

\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) + \frac{(\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{c} \times \mathbf{a})}{|\mathbf{b}|^2} \mathbf{b}

\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) + \frac{(\mathbf{a} \times \mathbf{c}) \cdot (\mathbf{a} \times \mathbf{b})}{|\mathbf{b}|^2} \mathbf{b}

\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) + \frac{(\mathbf{b} \times \mathbf{c}) \cdot (\mathbf{b} \times \mathbf{a})}{|\mathbf{b}|^2} \mathbf{b}

\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) + \frac{(\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{b} \times \mathbf{c})}{|\mathbf{b}|^2} \mathbf{b}