MathematicsChapter 3: Vectors
Addition and Geometry
Add vectors componentwise (or tip-to-tail / parallelogram).
Learning objectives
- State the main ideas and notation used in addition and geometry.
- Apply the rules of addition and geometry in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Definition
Add vectors componentwise (or tip-to-tail / parallelogram). Triangle law: a + b is the third side. Scalar multiply: stretch or shrink, reverse if the scalar is negative. Section formula / midpoint: midpoint of AB is a+b2 if a,b are position vectors. Meaning: addition is 'follow one arrow, then the other'.
Formula
AB = b - a
Where / symbols
- ADDITION AND GEOMETRY
- Add vectors componentwise (or tip-to-tail / parallelogram). Triangle law: a + b is the third side. Scalar multiply: stretch or shrink, reverse if the scalar is negative. Section formula / midpoint: midpoint of AB is (a+b)/2 if a,b are position vectors. Meaning: addition is 'follow one arrow, then the other'.
- 1. ADD
- <1,2,3>+<4,-1,0>=<5,1,3>. Parallelogram: diagonal is the sum. Subtract: a-b = a+(-b).
- 2. POINTS
- Vector AB = b - a (position of B minus A). Length |AB| is the distance. Parallel vectors: one is a scalar times the other (components in the same ratio).
Overview
Tip to tail, then points:
Add
<1,2,3>+<4,-1,0>=<5,1,3>. Parallelogram: diagonal is the sum. Subtract: a-b = a+(-b).
Examples Triangle law · Parallelogram diagonal · -b opposite
Points
Vector AB = b - a (position of B minus A). Length |AB| is the distance. Parallel vectors: one is a scalar times the other (components in the same ratio).
Examples Distance |b-a| · Midpoint · Parallel: a = k b
WHAT — ADDITION AND GEOMETRY
Add vectors componentwise (or tip-to-tail / parallelogram). Triangle law: a + b is the third side. Scalar multiply: stretch or shrink, reverse if the scalar is negative. Section formula / midpoint: midpoint of AB is a+b2 if a,b are position vectors. Meaning: addition is 'follow one arrow, then the other'.
RULE — 1. ADD
<1,2,3>+<4,-1,0>=<5,1,3>. Parallelogram: diagonal is the sum. Subtract: a-b = a+(-b).
METHOD — 2. POINTS
Vector AB = b - a (position of B minus A). Length |AB| is the distance. Parallel vectors: one is a scalar times the other (components in the same ratio).
WORKED EXAMPLE
Worked: AB if A is i+j and B is 4i - j + k. AB = b - a = (4i-j+k) - (i+j) = 3i - 2j + k
How to answer a addition and geometry 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
- AB if A is i+j and B is 4i - j + k.
- AB = b - a
- = (4i-j+k) - (i+j)
- = 3i - 2j + k
In simple words
AB is not a+b. It is 'to B, minus to A'.
Key points
- add matching components
- midpoint a+b2
- components
- AB = b-a
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Addition and Geometry
Sticky Notes · Quick Revision
Core Idea
- Add vectors componentwise (or tip-to-tail / parallelogram).
- Triangle law: a + b is the third side.
- Scalar multiply: stretch or shrink, reverse if the scalar is negative.
1. ADD
- <1,2,3>+<4,-1,0>=<5,1,3>.
- Parallelogram: diagonal is the sum.
- Subtract: a-b = a+(-b).
2. POINTS
- Vector AB = b - a (position of B minus A).
- Length |AB| is the distance.
- Parallel vectors: one is a scalar times the other (components in the same ratio).
Worked Example
- Worked: AB if A is i+j and B is 4i - j + k.
- AB = b - a
- = (4i-j+k) - (i+j)
- = 3i - 2j + k
Common Misconception
- AB is not a+b. It is 'to B, minus to A'.
FBISE Exam Check
- State the rule for addition and geometry before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- Add vectors componentwise (or tip-to-tail / parallelogram).
- Triangle law: a + b is the third side.
- Scalar multiply: stretch or shrink, reverse if the scalar is negative.
- AB is not a+b. It is 'to B, minus to A'.
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