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Triple (Scalar) Product

Triple (Scalar) Product

A way of multiplying three vectors in which the product is a scalar. The absolute value of a triple product is the volume of the parallelepiped formed by the three vectors.

The triple product of vectors u, v, and w is u·(v×w).

 

 

See also

Dot product, cross product

Key Formula

u⋅(v×w)=∣u1u2u3v1v2v3w1w2w3∣\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = \begin{vmatrix} u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \end{vmatrix}
Where:
  • u\mathbf{u} = First vector, with components (u₁, u₂, u₃)
  • v\mathbf{v} = Second vector, with components (v₁, v₂, v₃)
  • w\mathbf{w} = Third vector, with components (w₁, w₂, w₃)

Worked Example

Problem: Find the triple scalar product of u = (1, 2, 3), v = (4, 5, 6), and w = (7, 8, 0), and determine the volume of the parallelepiped they form.
Step 1: Set up the 3×3 determinant with u, v, and w as rows.
u⋅(v×w)=∣123456780∣\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = \begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 0 \end{vmatrix}
Step 2: Expand along the first row using cofactor expansion.
=1∣5680∣−2∣4670∣+3∣4578∣= 1\begin{vmatrix} 5 & 6 \\ 8 & 0 \end{vmatrix} - 2\begin{vmatrix} 4 & 6 \\ 7 & 0 \end{vmatrix} + 3\begin{vmatrix} 4 & 5 \\ 7 & 8 \end{vmatrix}
Step 3: Evaluate each 2×2 determinant.
=1(5⋅0−6⋅8)−2(4⋅0−6⋅7)+3(4⋅8−5⋅7)= 1(5 \cdot 0 - 6 \cdot 8) - 2(4 \cdot 0 - 6 \cdot 7) + 3(4 \cdot 8 - 5 \cdot 7)
Step 4: Simplify each term.
=1(−48)−2(−42)+3(−3)=−48+84−9=27= 1(-48) - 2(-42) + 3(-3) = -48 + 84 - 9 = 27
Step 5: The volume of the parallelepiped is the absolute value of this result.
V=∣27∣=27V = |27| = 27
Answer: The triple scalar product is 27, so the volume of the parallelepiped formed by u, v, and w is 27 cubic units.

Frequently Asked Questions

What does the sign of the triple scalar product tell you?
A positive triple scalar product means the three vectors u, v, w form a right-handed system (like your thumb, index finger, and middle finger on your right hand). A negative value means they form a left-handed system. If the product is zero, the three vectors are coplanar—they all lie in the same plane, so the parallelepiped has no volume.
Does the order of the vectors matter in the triple scalar product?
Cycling the vectors preserves the value: u · (v × w) = v · (w × u) = w · (u × v). However, swapping any two vectors flips the sign. For example, u · (w × v) = −u · (v × w). Since volume uses the absolute value, the order does not affect the volume but does affect the sign.

Triple Scalar Product vs. Triple Vector Product

The triple scalar product u · (v × w) produces a scalar and gives volume information. The triple vector product u × (v × w) produces a vector and can be expanded using the BAC-CAB identity: u × (v × w) = v(u · w) − w(u · v).

Why It Matters

The triple scalar product is the standard tool for computing the volume of a parallelepiped in 3D space. Dividing that volume by 6 gives the volume of the tetrahedron formed by the same three edge vectors. It also appears in physics when calculating magnetic flux and in determining whether three vectors are coplanar—a test that arises in linear algebra, computer graphics, and engineering.

Common Mistakes

Mistake: Computing the cross product of u and v first, then dotting with w, instead of following the given order u · (v × w).
Correction: Be careful about which two vectors are crossed. Crossing a different pair changes the sign. Stick to the order specified in the problem: first compute v × w, then dot the result with u.
Mistake: Forgetting to take the absolute value when finding volume.
Correction: The triple scalar product can be negative (indicating a left-handed orientation). Volume is always non-negative, so you must take |u · (v × w)| for the parallelepiped volume.

Related Terms

  • Vector — The objects combined in this product
  • Scalar — The type of quantity the product yields
  • Dot Product — Used as the outer operation on the cross product result
  • Cross Product — Used as the inner operation on two of the vectors
  • Parallelepiped — Its volume equals the absolute value of the product
  • Volume — Primary geometric quantity this product computes
  • Absolute Value — Applied to the product to obtain volume