MathematicsChapter 3: Vectors
Scalar Triple Product
The scalar triple product is a · (b x c), a single number.
Learning objectives
- State the main ideas and notation used in scalar triple product.
- Apply the rules of scalar triple product in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Definition
The scalar triple product is a · (b x c), a single number. Geometrically it is the signed volume of the parallelepiped on a, b, c. In components it is det with rows (or columns) the three vectors. a · (b x c) = 0 means coplanar (volume 0). Swapping two vectors changes sign. Cyclic a·(b x c)=b·(c x a)=c·(a x b). Meaning: three vectors, one volume.
Mathematical form
k = i i
i = 1 det with rows i
I = 1
Where / symbols
- SCALAR TRIPLE PRODUCT
- The scalar triple product is a · (b x c), a single number. Geometrically it is the signed volume of the parallelepiped on a, b, c. In components it is det with rows (or columns) the three vectors. a · (b x c) = 0 means coplanar (volume 0). Swapping two vectors changes sign. Cyclic a·(b x c)=b·(c x a)=c·(a x b). Meaning: three vectors, one volume.
- 1. VOLUME
- Absolute value is the volume. Sign: right-handed order positive (school). If two vectors parallel, volume 0. Units: product of three lengths.
- 2. DET FORM
- Write the 3x3 and expand. a · (b x c) = (a x b) · c same value. Do not write a x b x c without brackets (not associated the same way).
Relationship
- a·(b x c) = det[a b c]
- 0 => coplanar
Overview
Volume, then the determinant:
Volume
Absolute value is the volume. Sign: right-handed order positive (school). If two vectors parallel, volume 0. Units: product of three lengths.
Examples Parallelepiped · Coplanar: 0 · Sign from order
Det Form
Write the 3x3 and expand. a · (b x c) = (a x b) · c same value. Do not write a x b x c without brackets (not associated the same way).
Examples Rows = the three vectors · Expand any row · Check with a small example
WHAT — SCALAR TRIPLE PRODUCT
The scalar triple product is a · (b x c), a single number. Geometrically it is the signed volume of the parallelepiped on a, b, c. In components it is det with rows (or columns) the three vectors. a · (b x c) = 0 means coplanar (volume 0). Swapping two vectors changes sign. Cyclic a·(b x c)=b·(c x a)=c·(a x b). Meaning: three vectors, one volume.
RULE — 1. VOLUME
Absolute value is the volume. Sign: right-handed order positive (school). If two vectors parallel, volume 0. Units: product of three lengths.
METHOD — 2. DET FORM
Write the 3x3 and expand. a · (b x c) = (a x b) · c same value. Do not write a x b x c without brackets (not associated the same way).
WORKED EXAMPLE
Worked: i · (j x k). j x k = i i · i = 1 det with rows i,j,k is det I = 1
How to answer a scalar triple product question
- Identify the given quantities, notation, and required result.
- Choose the matching rule and write it before substitution.
- Work one algebraic step at a time, keeping brackets and signs clear.
- Check restrictions, simplify the result, and verify it where possible.
Worked example
- i · (j x k).
- j x k = i
- i · i = 1
- det with rows i,j,k is det I = 1
In simple words
Triple product is a scalar. Brackets: a · (b x c), not (a · b) x c (that is illegal types).
Key points
- a·(b x c) = det[a b c]
- 0 => coplanar
- |triple| = volume
- 3x3 det
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Scalar Triple Product
Sticky Notes · Quick Revision
Core Idea
- The scalar triple product is a · (b x c), a single number.
- Geometrically it is the signed volume of the parallelepiped on a, b, c.
- In components it is det with rows (or columns) the three vectors.
1. VOLUME
- Absolute value is the volume.
- Sign: right-handed order positive (school).
- If two vectors parallel, volume 0.
- Units: product of three lengths.
2. DET FORM
- Write the 3x3 and expand.
- a · (b x c) = (a x b) · c same value.
- Do not write a x b x c without brackets (not associated the same way).
Worked Example
- Worked: i · (j x k).
- j x k = i
- i · i = 1
- det with rows i,j,k is det I = 1
Common Misconception
- Triple product is a scalar. Brackets: a · (b x c), not (a · b) x c (that is illegal types).
FBISE Exam Check
- State the rule for scalar triple product before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- The scalar triple product is a · (b x c), a single number.
- Geometrically it is the signed volume of the parallelepiped on a, b, c.
- In components it is det with rows (or columns) the three vectors.
- Triple product is a scalar. Brackets: a · (b x c), not (a · b) x c (that is illegal types).
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