MathematicsChapter 3: Vectors
Scalars, Vectors, Components
A scalar has size only (temperature, mass, |v|).
Learning objectives
- State the main ideas and notation used in scalars, vectors, components.
- Apply the rules of scalars, vectors, components in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Definition
A scalar has size only (temperature, mass, |v|). A vector has size and direction (displacement, force, velocity). In space we write a = a1 i + a2 j + a3 k, with i, j, k unit vectors along x, y, z. Position vector of point P(x,y,z) is OP = x i + y j + z k. Meaning: the three numbers are the shadows on the three axes.
Mathematical form
a = a1 i + a2 j + a3 k
OP = x i + y j + z k. Meaning
hat = (26)i - (46)j + (46)k
Where / symbols
- SCALARS, VECTORS, COMPONENTS
- A scalar has size only (temperature, mass, |v|). A vector has size and direction (displacement, force, velocity). In space we write a = a1 i + a2 j + a3 k, with i, j, k unit vectors along x, y, z. Position vector of point P(x,y,z) is OP = x i + y j + z k. Meaning: the three numbers are the shadows on the three axes.
- 1. TELL APART
- Speed is scalar; velocity is vector. Distance vs displacement. Two vectors equal: same components. Zero vector: all components 0, no direction.
- 2. COMPONENTS
- a = <a1,a2,a3>. Magnitude |a| = sqrt(a1^2+a2^2+a3^2). Unit vector a-hat = a / |a| if a != 0. Negative -a flips direction.
Overview
What kind of quantity, then i j k:
Tell Apart
Speed is scalar; velocity is vector. Distance vs displacement. Two vectors equal: same components. Zero vector: all components 0, no direction.
Examples 5 m/s speed · 5 m/s east velocity · 0 has no dir
Components
a = <a1,a2,a3>. Magnitude |a| = sqrt(a1^2+a2^2+a3^2). Unit vector a-hat = a / |a| if a != 0. Negative -a flips direction.
Examples i = <1,0,0> · |3i+4j|=5 · unit = divide by length
WHAT — SCALARS, VECTORS, COMPONENTS
A scalar has size only (temperature, mass, |v|). A vector has size and direction (displacement, force, velocity). In space we write a = a1 i + a2 j + a3 k, with i, j, k unit vectors along x, y, z. Position vector of point P(x,y,z) is OP = x i + y j + z k. Meaning: the three numbers are the shadows on the three axes.
RULE — 1. TELL APART
Speed is scalar; velocity is vector. Distance vs displacement. Two vectors equal: same components. Zero vector: all components 0, no direction.
METHOD — 2. COMPONENTS
a = <a1,a2,a3>. Magnitude |a| = sqrt(a1^2+a2^2+a3^2). Unit vector a-hat = a / |a| if a != 0. Negative -a flips direction.
WORKED EXAMPLE
Worked: unit vector of 2 i - 4 j + 4 k. |a| = sqrt(4+16+16) = sqrt(36) = 6 a-hat = (26)i - (46)j + (46)k = (13)i - (23)j + (23)k
How to answer a scalars, vectors, components 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
- unit vector of 2 i - 4 j + 4 k.
- |a| = sqrt(4+16+16) = sqrt(36) = 6
- a-hat = (26)i - (46)j + (46)k
- = (13)i - (23)j + (23)k
In simple words
A unit vector has length 1, not 'length i'. Divide by |a|.
Key points
- scalar: size
- vector: size + direction
- arrow vs number
- three numbers
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Scalars, Vectors, Components
Sticky Notes · Quick Revision
Core Idea
- A scalar has size only (temperature, mass, |v|).
- A vector has size and direction (displacement, force, velocity).
- In space we write a = a1 i + a2 j + a3 k, with i, j, k unit vectors along x, y, z.
1. TELL APART
- Speed is scalar; velocity is vector.
- Distance vs displacement.
- Two vectors equal: same components.
- Zero vector: all components 0, no direction.
2. COMPONENTS
- a = <a1,a2,a3>.
- Magnitude |a| = sqrt(a12+a22+a32).
- Unit vector a-hat = a / |a| if a != 0.
- Negative -a flips direction.
Worked Example
- Worked: unit vector of 2 i - 4 j + 4 k.
- |a| = sqrt(4+16+16) = sqrt(36) = 6
- a-hat = (26)i - (46)j + (46)k
- = (13)i - (23)j + (23)k
Common Misconception
- A unit vector has length 1, not 'length i'. Divide by |a|.
FBISE Exam Check
- State the rule for scalars, vectors, components before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- A scalar has size only (temperature, mass, |v|).
- A vector has size and direction (displacement, force, velocity).
- In space we write a = a1 i + a2 j + a3 k, with i, j, k unit vectors along x, y, z.
- A unit vector has length 1, not 'length i'. Divide by |a|.
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