MathematicsChapter 3: Vectors
Exam Response (vectors)
Typical tasks: |a|; unit vector; AB=b-a; a·b and theta; a x b from the ijk det; area; a·(b x c) volume; perp vs parallel tests.
Learning objectives
- State the main ideas and notation used in exam response (vectors).
- Apply the rules of exam response (vectors) in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Definition
Typical tasks: |a|; unit vector; AB=b-a; a·b and theta; a x b from the ijk det; area; a·(b x c) volume; perp vs parallel tests. Show the 3x3 expansion with the minus on j. One component per line is fine.
Mathematical form
a=<1,2,-1>, b=<2,-1,0>
AB=b-a
b = 2 - 2 + 0
Where / symbols
- EXAM RESPONSE (VECTORS)
- Typical tasks: |a|; unit vector; AB=b-a; a·b and theta; a x b from the ijk det; area; a·(b x c) volume; perp vs parallel tests. Show the 3x3 expansion with the minus on j. One component per line is fine.
- 1. TESTS
- perp: dot 0. parallel: cross 0. coplanar: triple 0. unit: divide by |a|.
- 2. COMPUTE
- Dot: sum products. Cross: ijk det. Triple: 3x3 det of the three vectors.
Relationship
- dot = scalar, cross = vector
- AB = b-a
Overview
Tests, then a product:
Tests
perp: dot 0. parallel: cross 0. coplanar: triple 0. unit: divide by |a|.
Examples a·b=0 · a x b=0 · det=0
Compute
Dot: sum products. Cross: ijk det. Triple: 3x3 det of the three vectors.
Examples cos theta · area |cross| · volume |triple|
WHAT — EXAM RESPONSE (VECTORS)
Typical tasks: |a|; unit vector; AB=b-a; a·b and theta; a x b from the ijk det; area; a·(b x c) volume; perp vs parallel tests. Show the 3x3 expansion with the minus on j. One component per line is fine.
RULE — 1. TESTS
perp: dot 0. parallel: cross 0. coplanar: triple 0. unit: divide by |a|.
METHOD — 2. COMPUTE
Dot: sum products. Cross: ijk det. Triple: 3x3 det of the three vectors.
WORKED EXAMPLE
Worked 4-mark perp check. a=<1,2,-1>, b=<2,-1,0> a·b = 2 - 2 + 0 = 0 so a perp b (neither is 0)
How to answer a exam response (vectors) 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
- Worked 4-mark perp check.
- a=<1,2,-1>, b=<2,-1,0>
- a·b = 2 - 2 + 0 = 0
- so a perp b (neither is 0)
In simple words
Do not use a x b when the question asks for work (that is a dot).
Key points
- dot = scalar, cross = vector
- AB = b-a
- zero tests
- write the matrix
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Exam Response (vectors)
Sticky Notes · Quick Revision
Core Idea
- Typical tasks: |a|; unit vector; AB=b-a; a·b and theta; a x b from the ijk det; area; a·(b x c) volume; perp vs parallel tests.
- Show the 3x3 expansion with the minus on j.
- One component per line is fine.
1. TESTS
- perp: dot 0.
- parallel: cross 0.
- coplanar: triple 0.
- unit: divide by |a|.
2. COMPUTE
- Dot: sum products.
- Cross: ijk det.
- Triple: 3x3 det of the three vectors.
Worked Example
- Worked 4-mark perp check.
- a=<1,2,-1>, b=<2,-1,0>
- a·b = 2 - 2 + 0 = 0
- so a perp b (neither is 0)
Common Misconception
- Do not use a x b when the question asks for work (that is a dot).
FBISE Exam Check
- State the rule for exam response (vectors) before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- Typical tasks: |a|; unit vector; AB=b-a; a·b and theta; a x b from the ijk det; area; a·(b x c) volume; perp vs parallel tests.
- Show the 3x3 expansion with the minus on j.
- One component per line is fine.
- Do not use a x b when the question asks for work (that is a dot).
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