Scalar Triple Product, Coplanarity, and Volume

For $\vec{a} = a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$, $\vec{b} = b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$, $\vec{c} = c_1\hat{i}+c_2\hat{j}+c_3\hat{k}$:

• Notation: $[\vec{a}\ \vec{b}\ \vec{c}] = \vec{a}\cdot(\vec{b}\times\vec{c})$ — a single scalar
• Determinant form (the workhorse): $[\vec{a}\ \vec{b}\ \vec{c}] = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}$
• Dot-cross interchange: $\vec{a}\cdot(\vec{b}\times\vec{c}) = (\vec{a}\times\vec{b})\cdot\vec{c}$ — the dot and cross can swap places without changing the value
• Geometric meaning: $[\vec{a}\ \vec{b}\ \vec{c}]$ is the SIGNED volume of the parallelepiped with edge vectors $\vec{a}, \vec{b}, \vec{c}$

For any $\vec{a}, \vec{b}, \vec{c}$:

• Cyclic (rotation is free): $[\vec{a}\ \vec{b}\ \vec{c}] = [\vec{b}\ \vec{c}\ \vec{a}] = [\vec{c}\ \vec{a}\ \vec{b}]$
• Swap flips the sign: $[\vec{a}\ \vec{b}\ \vec{c}] = -[\vec{b}\ \vec{a}\ \vec{c}] = -[\vec{a}\ \vec{c}\ \vec{b}]$
• Repeated vector ⇒ zero: $[\vec{a}\ \vec{a}\ \vec{b}] = 0$ (two equal rows make the determinant vanish)
• Linearity in each slot: $[\alpha\vec{u}+\beta\vec{v}\ \ \vec{b}\ \ \vec{c}] = \alpha[\vec{u}\ \vec{b}\ \vec{c}] + \beta[\vec{v}\ \vec{b}\ \vec{c}]$

• Perpendicular case: if $\vec{a}$ is perpendicular to both $\vec{b}$ and $\vec{c}$, then $\vec{a}$ is parallel to $\vec{b}\times\vec{c}$, so $[\vec{a}\ \vec{b}\ \vec{c}] = \pm|\vec{a}|\,|\vec{b}\times\vec{c}| = \pm|\vec{a}|\,|\vec{b}||\vec{c}|\sin\theta$ where $\theta$ is the angle between $\vec{b}$ and $\vec{c}$.
• Independence: if the determinant of the component matrix reduces to a constant, $[\vec{a}\ \vec{b}\ \vec{c}]$ does not depend on the parameters inside the vectors.
• Magnitude relations: a datum like $|\vec{b}\times\vec{c}|$ fixes $\sin\theta$, and a relation $\vec{b} = 2\vec{c} + \lambda\vec{a}$ is solved by taking magnitudes: $|\vec{b} - 2\vec{c}|^2 = \lambda^2|\vec{a}|^2$.

$\vec{a}, \vec{b}, \vec{c}$ are coplanar $\iff [\vec{a}\ \vec{b}\ \vec{c}] = 0$, i.e. the determinant of their components is zero. If $\vec{c}$ lies in the plane of $\vec{a}$ and $\vec{b}$, then $[\vec{a}\ \vec{b}\ \vec{c}] = 0$ too. To find a parameter $\lambda$ (or $x$) that makes them coplanar, set the determinant equal to zero and solve the resulting equation; a parameter that appears squared can give two distinct real values.

Expanding by linearity and deleting repeated-vector terms:

• $[\vec{a}+\vec{b}\ \ \vec{b}+\vec{c}\ \ \vec{c}+\vec{a}] = 2[\vec{a}\ \vec{b}\ \vec{c}]$
• $[m\vec{a}+\vec{b}\ \ m\vec{b}+\vec{c}\ \ m\vec{c}+\vec{a}] = (m^3 + 1)[\vec{a}\ \vec{b}\ \vec{c}]$
• In general the new STP equals $\det(M)\,[\vec{a}\ \vec{b}\ \vec{c}]$, where $M$ is the 3x3 matrix of the combination coefficients.

If $\vec{a}, \vec{b}, \vec{c}$ are coplanar, $[\vec{a}\ \vec{b}\ \vec{c}] = 0$ so every such combination is also zero.

For edge vectors $\vec{a}, \vec{b}, \vec{c}$:

• Volume $= |[\vec{a}\ \vec{b}\ \vec{c}]| = \left|\det\text{(component matrix)}\right|$.
• Find a component: set $|[\vec{a}\ \vec{b}\ \vec{c}]| = V$ (given) and solve.
• Min/max in a parameter: write the determinant as a function $V(m)$, set $V'(m) = 0$, and use the sign of $V''(m)$ to classify minimum vs maximum.
• The dihedral angle between two faces of a tetrahedron is the angle between the two face normals, each found as a cross product of edge vectors.

For a tetrahedron with vertices $A, B, C, D$, form the edge vectors from one vertex: $\overrightarrow{AB}, \overrightarrow{AC}, \overrightarrow{AD}$. Then Volume $= \dfrac{1}{6}\,|[\overrightarrow{AB}\ \overrightarrow{AC}\ \overrightarrow{AD}]|$. Setting this equal to a given volume yields an equation for an unknown coordinate.

For non-coplanar $\vec{a}, \vec{b}, \vec{c}$, set $\vec{p} = \dfrac{\vec{b}\times\vec{c}}{[\vec{a}\ \vec{b}\ \vec{c}]}$, $\vec{q} = \dfrac{\vec{c}\times\vec{a}}{[\vec{a}\ \vec{b}\ \vec{c}]}$, $\vec{r} = \dfrac{\vec{a}\times\vec{b}}{[\vec{a}\ \vec{b}\ \vec{c}]}$:

• Matched pairs: $\vec{a}\cdot\vec{p} = \vec{b}\cdot\vec{q} = \vec{c}\cdot\vec{r} = 1$
• Mismatched pairs: $\vec{a}\cdot\vec{q} = \vec{a}\cdot\vec{r} = \vec{b}\cdot\vec{p} = \dots = 0$ (the cross product is perpendicular to its own factors)
• STP-squared rule: $[\vec{a}\times\vec{b}\ \ \vec{b}\times\vec{c}\ \ \vec{c}\times\vec{a}] = [\vec{a}\ \vec{b}\ \vec{c}]^2$

For any $\vec{a}, \vec{b}, \vec{c}$:

• $\vec{a}\times(\vec{b}\times\vec{c}) = (\vec{a}\cdot\vec{c})\,\vec{b} - (\vec{a}\cdot\vec{b})\,\vec{c}$
• $(\vec{a}\times\vec{b})\times\vec{c} = (\vec{a}\cdot\vec{c})\,\vec{b} - (\vec{b}\cdot\vec{c})\,\vec{a}$
• The vector triple product is NOT associative: $\vec{a}\times(\vec{b}\times\vec{c}) \neq (\vec{a}\times\vec{b})\times\vec{c}$ in general — the two lie in different planes.

Equating a given expansion to a stated multiple of $\vec{b}$ and $\vec{c}$ lets you read off $\vec{a}\cdot\vec{b}$ (hence the angle), since $\vec{b}, \vec{c}$ are independent.