(a) The triple cross product … is in the plane of …

(a) The triple cross product $(\mathbf{A}\times \mathbf{B})\times \mathbf{C}$ is in the plane of $\mathbf{A}$ and $\mathbf{B}$, because it is perpendicular to the cross product ______.
(b) Compute $(\mathbf{A}\times \mathbf{B})\times \mathbf{C}$ when $\mathbf{A}=a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k}$, $\mathbf{B}=b_1\mathbf{i}+b_2\mathbf{j}+b_3\mathbf{k}$, $\mathbf{C}=\mathbf{i}$.
(c) Compute $(\mathbf{A}\cdot \mathbf{C})\mathbf{B}-(\mathbf{B}\cdot \mathbf{C})\mathbf{A}$ when $\mathbf{C}=\mathbf{i}$. The answers in (b) and (c) should agree. This is also true if $\mathbf{C}=\mathbf{j}$ or $\mathbf{C}=\mathbf{k}$ or $\mathbf{C}=c_1\mathbf{i}+c_2\mathbf{j}+c_3\mathbf{k}$. That proves the tricky formula
$$(\mathbf{A}\times \mathbf{B})\times \mathbf{C}=(\mathbf{A}\cdot \mathbf{C})\mathbf{B}-(\mathbf{B}\cdot \mathbf{C})\mathbf{A}.(\ast )$$

Its source prints no answer for this problem, so there's nothing to check against.

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §11.3 Cross Products and Determinants, #47