Question

If $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are three vectors, the vector triple product $(\mathbf{a}\times\mathbf{b})\times\mathbf{c}$ is given by

• A. $(\mathbf{a}\cdot\mathbf{c})\mathbf{b} - (\mathbf{a}\cdot\mathbf{b})\mathbf{c}$
• B. $(\mathbf{a}\cdot\mathbf{b})\mathbf{c} - (\mathbf{a}\cdot\mathbf{c})\mathbf{b}$
• C. $(\mathbf{a}\cdot\mathbf{c})\mathbf{b} - (\mathbf{b}\cdot\mathbf{c})\mathbf{a}$
• D. $(\mathbf{b}\cdot\mathbf{c})\mathbf{a} - (\mathbf{a}\cdot\mathbf{c})\mathbf{b}$

Hint:

Remember: $(\mathbf{a}\times\mathbf{b})\times\mathbf{c}$ never contains $\mathbf{c}$ in the final result—it always reduces to a combination of $\mathbf{a}$ and $\mathbf{b}$.

Answer 1

The Correct Option is C

The identity for the vector triple product $(\mathbf{a} \times \mathbf{b}) \times \mathbf{c}$ is a standard vector identity.
It is given by:
\[ (\mathbf{a}\times\mathbf{b})\times\mathbf{c} = (\mathbf{a}\cdot\mathbf{c})\mathbf{b} - (\mathbf{b}\cdot\mathbf{c})\mathbf{a} \]
This identity shows that the result is a linear combination of vectors $\mathbf{a}$ and $\mathbf{b}$, weighted by dot products involving $\mathbf{c}$.
Comparing with the given options, option (C) exactly matches this identity.
Therefore, the correct answer is option (C).