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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.
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Scalar Triple Product
What is it?
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.
In simple words
- 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).
Important terms
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).
What you should learn
- 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.
Examples
- Parallelepiped
- Coplanar: 0
- Sign from order
VOLUME
Absolute value is the volume. Sign: right-handed order positive (school). If two vectors parallel, volume 0. Units: product of three lengths.
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).
Recommended for Class 11
Full school-level explanation
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.
Volume, then the determinant:
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
VOLUME
Absolute value is the volume. Sign: right-handed order positive (school). If two vectors parallel, volume 0. Units: product of three lengths.
Remember
- |triple| = volume
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).
Remember
- 3x3 det
Examples
- Rows = the three vectors
- Expand any row
- Check with a small example
How to answer a scalar triple product question
How it works
- 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.
Remember
- a·(b x c) = det[a b c]
- 0 => coplanar
- 1
- dot after cross
- |triple| = volume
- 3x3 det
- a·(b x c) = det[a b c]
- 0 => coplanar
How the ideas connect
Worked: i · (j x k). j x k = i i · i = 1 det with rows i,j,k is det I = 1
Deeper understanding
Go beyond memorising Scalar Triple Product. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: i · (j x k). j x k = i i · i = 1 det with rows i,j,k is det I = 1
Common mistakes
Exam focus
Core Idea
1. VOLUME
2. DET FORM
Worked Example
Exam focus
Quick revision
- a·(b x c) = det[a b c]
- 0 => coplanar
- 1
- dot after cross
- SCALAR TRIPLE PRODUCT: The scalar triple product is a · (b x c), a single number. Geometrically it is the signed volume of
- 1. VOLUME: Absolute value is the volume. Sign: right-handed order positive (school). If two vectors parallel, v
Triple product is a scalar. Brackets: a · (b x c), not (a · b) x c (that is illegal types).
For long answers on Scalar Triple Product: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- a·(b x c) = det[a b c]
- 0 => coplanar
- 1
- dot after cross
Important formulas
- a·(b x c) = det[a b c]
- 0 => coplanar
- 1
Common exam traps
- Triple product is a scalar. Brackets: a · (b x c), not (a · b) x c (that is illegal types).
- Do not use a remembered formula before checking that its conditions fit the question.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Triple product is a scalar. Brackets: a · (b x c), not (a · b) x c (that is illegal types).
- Mistake
Do not use a remembered formula before checking that its conditions fit the question.
- Mistake
Do not hide sign changes or restrictions; write them on the line where they are used.
- Mistake
Triple product is a scalar. Brackets: a · (b x c), not (a · b) x c (that is illegal types).
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
a·(b x c) = det[a b c]
0 => coplanar
1
dot after cross
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
a·(b x c) = det[a b c]
- 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
- Formula
- a·(b x c) = det[a b c]
- Example
- Parallelepiped
- Critical warning
- Triple product is a scalar. Brackets: a · (b x c), not (a · b) x c (that is illegal types).
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Exam Practice
Define
Q1. Define Scalar Triple Product.
Explain
Q2. Explain: VOLUME.
Example
Q3. Give one clear example of Scalar Triple Product and say why it fits.
Mistake
Q4. State one common mistake students make about Scalar Triple Product.
Exam note
Q5. Write a short exam note on Scalar Triple Product: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.