Scalar Triple Product
Volume of the parallelepiped spanned by three vectors.
Formula
Variables
Example
Zero means coplanar
How it works
The scalar triple product a·(b×c) measures how much of vector a points in the direction perpendicular to the parallelogram formed by b and c, so its absolute value gives the 3D volume spanned by the three vectors. The cross product b×c creates an area vector whose magnitude is the base area and whose direction is normal to the base; dotting with a picks out the corresponding height component. The sign tells orientation: reversing the order of two vectors changes the sign, but not the volume.
When to use it
Use it when you need the volume of a parallelepiped, or to test whether three vectors are coplanar by checking whether the result is zero. It also appears in coordinate geometry and physics whenever orientation, handedness, or mixed products of vectors matter.
Worked example, step by step
Let a = (2, 1, 3), b = (1, 0, 2), c = (3, -1, 1) Step 1: Compute b × c b × c = |i j k| |1 0 2| |3 -1 1| = (0·1 - 2·(-1), -(1·1 - 2·3), 1·(-1) - 0·3) = (2, 5, -1) Step 2: Dot with a a · (b × c) = (2, 1, 3) · (2, 5, -1) = 2·2 + 1·5 + 3·(-1) = 4 + 5 - 3 = 6 Step 3: Interpret the result Scalar triple product = 6 Volume of the parallelepiped = |6| = 6 cubic units
Common mistakes to avoid
- Forgetting to take the absolute value when the question asks for volume
- Computing the cross product in the wrong order, which changes the sign
- Mixing up the scalar triple product with the vector triple product a×(b×c)
Did You Know?
The scalar triple product gives the volume of a parallelepiped and is zero when three vectors lie in one plane.
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