Vector Product and Scalar Triple Product
The vector product a x b is perpendicular to both with magnitude |a||b| sin t, and the scalar triple product [a b c] gives the volume of the parallelepiped.
What is Vector Product and Scalar Triple Product?
The vector product a x b is perpendicular to both with magnitude |a||b| sin t, and the scalar triple product [a b c] gives the volume of the parallelepiped.
Key formula / rule: |a x b| is the area of the parallelogram spanned by a and b.
Key points
- Compute the vector product and interpret its magnitude as an area.
- Evaluate coplanarity of three vectors using the scalar triple product.
Common exam trap
Assuming the cross product is commutative.
Definitions
- Term
Vector Product and Scalar Triple Product
- Meaning
The vector product a x b is perpendicular to both with magnitude |a||b| sin t, and the scalar triple product [a b c] gives the volume of the parallelepiped.
- Term
Vector Product and Scalar Triple Product — explanation
- Meaning
The cross product encodes area and orientation, while the triple product tests coplanarity by vanishing. Anti-commutativity is the property most often forgotten.
Learning objectives
Compute the vector product and interpret its magnitude as an area.
Evaluate coplanarity of three vectors using the scalar triple product.
Formulae
- Key point
|a x b| is the area of the parallelogram spanned by a and b.
- Key point
a x b = -(b x a), so order matters.
- Key point
[a b c] = 0 exactly when the three vectors are coplanar.
Prerequisites
MAT-U4-VA-T1-S1-C2
Common mistakes
Assuming the cross product is commutative.
Using the triple product as a vector quantity.
Keywords
Vector
Product
Scalar
Triple
Practice preview
What is the magnitude of the vector product of two vectors a and b, where theta is the angle between them?…
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If i, j, k are unit vectors along the x, y, and z axes respectively, what is the value of i x j?…
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Find the volume of the parallelepiped whose co-terminus edges are given by the vectors a = i - 2j + 3k, b = 2i + j - k, and c = i + 3j - 2k.…
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