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Cross Product in Components
a x b = det of a 3x3 with row 1 = i j k, row 2 = components of a, row 3 = components of b (school mnemonic). Expand: (a2 b3 - a3 b2) i - (a1 b3 - a3 b1) j + (a1 b2 - a2 b1) k. Distributive: a x (b+c) = a x b + a x c. Angle: sin theta = |a x b| / (|a||b|). Meaning: the determinant is just a compact way to write those three 2x2 minors.
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Cross Product in Components
What is it?
a x b = det of a 3x3 with row 1 = i j k, row 2 = components of a, row 3 = components of b (school mnemonic). Expand: (a2 b3 - a3 b2) i - (a1 b3 - a3 b1) j + (a1 b2 - a2 b1) k. Distributive: a x (b+c) = a x b + a x c. Angle: sin theta = |a x b| / (|a||b|). Meaning: the determinant is just a compact way to write those three 2x2 minors.
In simple words
- a x b = det of a 3x3 with row 1 = i j k, row 2 = components of a, row 3 = components of b (school mnemonic).
- Expand: (a2 b3 - a3 b2) i - (a1 b3 - a3 b1) j + (a1 b2 - a2 b1) k.
- Distributive: a x (b+c) = a x b + a x c.
- The j-component formula already includes the minus from the cofactor. Do not minus twice.
Important terms
CROSS PRODUCT IN COMPONENTS: a x b = det of a 3x3 with row 1 = i j k, row 2 = components of a, row 3 = components of b (school mnemonic). Expand: (a2 b3 - a3 b2) i - (a1 b3 - a3 b1) j + (a1 b2 - a2 b1) k. Distributive: a x (b+c) = a x b + a x c. Angle: sin theta = |a x b| / (|a||b|). Meaning: the determinant is just a compact way to write those three 2x2 minors.
1. THE DETERMINANT: Write i j k on top. Compute the three cofactors. Watch the minus in front of j. Practice one fully written expansion.
2. USES: Area. Torque tau = r x F named (physics). Perp vector to a plane of two vectors. If the result is 0, the vectors are parallel (or one is 0).
What you should learn
- State the main ideas and notation used in cross product in components.
- Apply the rules of cross product in components in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Examples
- i (a2b3-a3b2)
- - j (a1b3-a3b1)
- k (a1b2-a2b1)
THE DETERMINANT
Write i j k on top. Compute the three cofactors. Watch the minus in front of j. Practice one fully written expansion.
USES
Area. Torque tau = r x F named (physics). Perp vector to a plane of two vectors. If the result is 0, the vectors are parallel (or one is 0).
Recommended for Class 11
Full school-level explanation
a x b = det of a 3x3 with row 1 = i j k, row 2 = components of a, row 3 = components of b (school mnemonic). Expand: (a2 b3 - a3 b2) i - (a1 b3 - a3 b1) j + (a1 b2 - a2 b1) k. Distributive: a x (b+c) = a x b + a x c. Angle: sin theta = |a x b| / (|a||b|). Meaning: the determinant is just a compact way to write those three 2x2 minors.
Expand, then the angle:
WHAT — CROSS PRODUCT IN COMPONENTS
a x b = det of a 3x3 with row 1 = i j k, row 2 = components of a, row 3 = components of b (school mnemonic). Expand: (a2 b3 - a3 b2) i - (a1 b3 - a3 b1) j + (a1 b2 - a2 b1) k. Distributive: a x (b+c) = a x b + a x c. Angle: sin theta = |a x b| / (|a||b|). Meaning: the determinant is just a compact way to write those three 2x2 minors.
RULE — 1. THE DETERMINANT
Write i j k on top. Compute the three cofactors. Watch the minus in front of j. Practice one fully written expansion.
METHOD — 2. USES
Area. Torque tau = r x F named (physics). Perp vector to a plane of two vectors. If the result is 0, the vectors are parallel (or one is 0).
WORKED EXAMPLE
Worked: <1,2,3> x <4,5,6>. i (12-15) - j (6-12) + k (5-8) i (-3) - j (-6) + k (-3) = -3 i + 6 j - 3 k
THE DETERMINANT
Write i j k on top. Compute the three cofactors. Watch the minus in front of j. Practice one fully written expansion.
Remember
- minus on j
Examples
- i (a2b3-a3b2)
- - j (a1b3-a3b1)
- k (a1b2-a2b1)
USES
Area. Torque tau = r x F named (physics). Perp vector to a plane of two vectors. If the result is 0, the vectors are parallel (or one is 0).
Remember
- area and normal
Examples
- sin theta from |cross|
- Check a · (a x b) = 0
- Distributive
How to answer a cross product in components 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
- ijk determinant mnemonic
- middle component has a minus
- <-3,6,-3>
- one minus on j
- minus on j
- area and normal
- ijk determinant mnemonic
- middle component has a minus
How the ideas connect
Worked: <1,2,3> x <4,5,6>. i (12-15) - j (6-12) + k (5-8) i (-3) - j (-6) + k (-3) = -3 i + 6 j - 3 k
Deeper understanding
Go beyond memorising Cross Product in Components. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: <1,2,3> x <4,5,6>. i (12-15) - j (6-12) + k (5-8) i (-3) - j (-6) + k (-3) = -3 i + 6 j - 3 k
Common mistakes
Exam focus
Core Idea
1. THE DETERMINANT
2. USES
Worked Example
Exam focus
Quick revision
- ijk determinant mnemonic
- middle component has a minus
- <-3,6,-3>
- one minus on j
- CROSS PRODUCT IN COMPONENTS: a x b = det of a 3x3 with row 1 = i j k, row 2 = components of a, row 3 = components of b (school mn
- 1. THE DETERMINANT: Write i j k on top. Compute the three cofactors. Watch the minus in front of j. Practice one fully w
The j-component formula already includes the minus from the cofactor. Do not minus twice.
For long answers on Cross Product in Components: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- ijk determinant mnemonic
- middle component has a minus
- <-3,6,-3>
- one minus on j
Important formulas
- ijk determinant mnemonic
- middle component has a minus
- <-3,6,-3>
Common exam traps
- The j-component formula already includes the minus from the cofactor. Do not minus twice.
- 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
The j-component formula already includes the minus from the cofactor. Do not minus twice.
- 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
The j-component formula already includes the minus from the cofactor. Do not minus twice.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
ijk determinant mnemonic
middle component has a minus
<-3,6,-3>
one minus on j
a x b = det of a 3x3 with row 1 = i j k, row 2 = components of a, row 3 = components of b (school mnemonic). Expand: (a2 b3 - a3 b2) i - (a1 b3 - a3 b1) j + (a1
- Definition
- a x b = det of a 3x3 with row 1 = i j k, row 2 = components of a, row 3 = components of b (school mnemonic). Expand: (a2 b3 - a3 b2) i - (a1 b3 - a3 b1) j + (a1
- Example
- i (a2b3-a3b2)
- Critical warning
- The j-component formula already includes the minus from the cofactor. Do not minus twice.
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Exam Practice
Define
Q1. Define Cross Product in Components.
Explain
Q2. Explain: THE DETERMINANT.
Example
Q3. Give one clear example of Cross Product in Components and say why it fits.
Mistake
Q4. State one common mistake students make about Cross Product in Components.
Exam note
Q5. Write a short exam note on Cross Product in Components: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.