If $\vec a,\vec b,\vec c$ are three non-coplanar vectors represented by concurrent edges of a parallelepiped of volume $4$ cubic units, find the value of $(\vec a+\vec b)\cdot(\vec b imes\vec c)+(\vec b+\vec c)\cdot(\vec c imes\vec a)+(\vec c+\vec a)\cdot(\vec a imes\vec b)$.

Question

If $\vec a,\vec b,\vec c$ are three non-coplanar vectors represented by concurrent edges of a parallelepiped of volume $4$ cubic units, find the value of $(\vec a+\vec b)\cdot(\vec b	imes\vec c)+(\vec b+\vec c)\cdot(\vec c	imes\vec a)+(\vec c+\vec a)\cdot(\vec a	imes\vec b)$.

Accepted Answer

> [!TLDR]
>
> Expand the sum of three dot products term-by-term; every "self" term like $\vec b\cdot(\vec b	imes\vec c)$ vanishes automatically (a vector dotted with something perpendicular to it), leaving three cyclic copies of the same scalar triple product.

**Step 1. Expand each bracket.**
$$(\vec a+\vec b)\cdot(\vec b	imes\vec c)=\vec a\cdot(\vec b	imes\vec c)+\vec b\cdot(\vec b	imes\vec c).$$
$$(\vec b+\vec c)\cdot(\vec c	imes\vec a)=\vec b\cdot(\vec c	imes\vec a)+\vec c\cdot(\vec c	imes\vec a).$$
$$(\vec c+\vec a)\cdot(\vec a	imes\vec b)=\vec c\cdot(\vec a	imes\vec b)+\vec a\cdot(\vec a	imes\vec b).$$

**Step 2. Kill the "self" terms.** $\vec b	imes\vec c\perp\vec b$, so $\vec b\cdot(\vec b	imes\vec c)=0$; likewise $\vec c\cdot(\vec c	imes\vec a)=0$ and $\vec a\cdot(\vec a	imes\vec b)=0$.

**Step 3. What remains.** …

Q.If $\vec a,\vec b,\vec c$ are three non-coplanar vectors represented by concurrent edges of a parallelepiped of volume $4$ cubic units, find the value of $(\vec a+\vec b)\cdot(\vec b\times\vec c)+(\vec b+\vec c)\cdot(\vec c\times\vec a)+(\vec c+\vec a)\cdot(\vec a\times\vec b)$ .

Concept understanding — Scalar Triple Product and Coplanarity

Scalar Triple Product and Coplanarity

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