Vector triple product

The vector triple product is defined as the cross product of one vector with the cross product of the other two. Consider the triple product $\boldsymbol{a} \times(\boldsymbol{b} \times \boldsymbol{c})$; note that the brackets are important since the vector product is anticommutative. We observe that $\boldsymbol{b} \times \boldsymbol{c}$ gives a vector that is perpendicular to the plane containing $\boldsymbol{b}$ and $\boldsymbol{c}$. Taking the cross product of $\boldsymbol{a}$ must then put us back in the plane described by $\boldsymbol{b}$ and $\boldsymbol{c}$. This gives us

where $\lambda$ and $\mu$ are given by the following relationship,

This is known as the triple product expansion whose RHS can be remembered using the mnemonic 'BAC-CAB'. Since the formula is anticommutative, it can also be written as

Exercises

1. Test the points $(0,1,1)(1,2,2)(3,2,2)$, and (9.6.6) for coplanarity.
2. Do the vectors $(1,1,2),(1,3,1)$ and $(1,-3,4)$ lie on the same plane through the origin? If they do, express one vector as a linear combination of the other two.
3. Give the equation of the plane in both Cartesian and parametric form containing the vectors $(1,0,1)(2,2,2)$ and $(-1,2,3)$.