➡️ Vectors
Forces, velocities and displacements all have both size and direction — vectors are the mathematical language for handling both together, from resolving forces to calculating work done.
2Topics
8Worked Examples
4Eng. Cases
12Practice Qs
➡️Representation & AdditionComponents, magnitude, direction
✖️Scalar & Vector ProductsDot and cross products
⚙️ApplicationsReal engineering
🎮Quick-Fire GameTest your speed
📝PracticeGraded questions
🎯What you will learn
Represent vectors in component (i,j) form; add, subtract and find the magnitude and direction of vectors; and calculate the scalar (dot) product and vector (cross) product, with their direct engineering meanings.
💡 The scalar product gives a single number related to how aligned two vectors are (used for work done); the vector product gives a new vector perpendicular to both (used for moments and torque).
🧭How to use this resource
1. Read the rules banner at the top of each tab
2. Experiment with the interactive tool
3. Check the key vocabulary panel
4. Study the visual diagram, then the worked examples
➡️
Vector Representation & Addition
Every vector can be broken into perpendicular components — the key that makes addition simple.
🔧Vector Magnitude & Direction CalculatorEnter x and y components
Enter x and y components, then press Calculate to find the magnitude and direction…
📖 Key Vocabulary
Scalar — a quantity with magnitude only (e.g. mass, temperature).
Vector — a quantity with both magnitude and direction (e.g. force, velocity).
Unit vectors i, j — vectors of length 1 along the x- and y-axes, used to build any 2D vector.
Resultant — the single vector equivalent to the sum of two or more vectors.
👁️ Visual aid: the resultant a+b runs directly from the start of a to the tip of b — component addition gives exactly the same answer, just algebraically.
✏️Worked Example 1 — Finding magnitude and directionType: Magnitude/direction
Question Find the magnitude and direction of v = 6i + 8j.
Magnitude
|v| = √(6²+8²) = √100 = 10Direction
θ = tan⁻¹(8/6) = 53.13°Answer |v|=10, θ=53.13° from the positive x-axis
➕Worked Example 2 — Adding two vectors in component formType: Addition
Question Add a = 3i + 5j and b = 7i − 2j.
Add components
x: 3+7=10, y: 5+(−2)=3Answer a+b = 10i + 3j
➖Worked Example 3 — Subtracting vectorsType: Subtraction
Question Find a − b for a = 4i + 9j, b = 6i + 2j.
Subtract components
x: 4−6=−2, y: 9−2=7Answer a−b = −2i + 7j
⚙️Worked Example 4 — Engineering context (resolving forces)Type: Engineering
Question Two forces act on a bracket: F₁=(50, 20)N and F₂=(−15, 35)N. Find the resultant force's magnitude and direction.
Add components
x: 50+(−15)=35, y: 20+35=55Magnitude
√(35²+55²) = √(1225+3025) = √4250 ≈ 65.19NAnswer Resultant ≈ 65.19N at θ=tan⁻¹(55/35)=57.53° — the single equivalent force the bracket actually experiences.
🧠 Common slip: adding magnitudes directly without considering direction — vectors must be added component-by-component (or tip-to-tail graphically), never by simply summing their lengths.
✖️
Scalar & Vector Products
Two different ways to "multiply" vectors, each with a distinct physical meaning.
🔧Dot & Cross Product CalculatorEnter two vectors in component form
Enter both vectors' components, then press Calculate for the dot product, cross product, and angle between them…
📖 Key Vocabulary
Scalar (dot) product — a·b, giving a single number; used for work done = F·d.
Vector (cross) product — a×b, giving a vector perpendicular to both; used for moments and torque.
Orthogonal vectors — perpendicular vectors, where a·b = 0.
Parallel vectors — where a×b = 0 (in 2D, the cross product magnitude is zero).
👁️ Visual aid: same two input vectors, but the two products answer fundamentally different physical questions.
✏️Worked Example 1 — Calculating the dot productType: Dot product
Question Find a·b for a=3i+4j, b=5i+12j.
Substitute
a·b = (3)(5) + (4)(12)Answer a·b = 15+48 = 63
📐Worked Example 2 — Finding the angle between vectorsType: Angle via dot product
Question Using a=3i+4j (|a|=5) and b=5i+12j (|b|=13), a·b=63 (from Example 1), find the angle between them.
Rearrange
cosθ = a·b/(|a||b|) = 63/(5×13) = 63/65Answer θ = cos⁻¹(0.9692) ≈ 14.25°
✖️Worked Example 3 — Calculating the cross product (2D)Type: Cross product
Question Find the magnitude of a×b for a=3i+4j, b=5i+12j.
Substitute
a×b = (3)(12) − (4)(5) = 36−20Answer a×b = 16 (this scalar in 2D represents the z-component of the full 3D cross product)
⚙️Worked Example 4 — Engineering context (work done by a force)Type: Engineering
Question A force F=(40, 30)N moves an object through displacement d=(6, 2)m. Find the work done.
Apply formula
W = F·d = (40)(6) + (30)(2)Answer W = 240+60 = 300 J — only the component of force along the direction of displacement contributes to work, which is exactly what the dot product captures.
🧠 Common slip: using the cross product formula when a scalar answer (like work done) is needed, or vice versa — always match the product type to what physical quantity the question is actually asking for.
⚙️
Engineering Applications
Four live scenarios — each with its own calculator. Enter your own numbers.
🔧 Resultant Force on a Structure
Combining multiple force vectors acting on a joint or bracket into a single equivalent resultant.
Resultant = Σx i + Σy j
📟 Try it — Resultant Force Calculator
Enter both force vectors, then press Calculate…
⚡ Work Done by a Force
The dot product of force and displacement gives the work done directly — a core mechanics calculation.
W = F·d
📟 Try it — Work Done Calculator
Enter force and displacement vectors, then press Calculate…
🔩 Moment of a Force (Torque)
The cross product of a position vector and a force gives the moment (turning effect) about a point.
M = r × F
📟 Try it — Moment Calculator
Enter position and force vectors, then press Calculate…
🚗 Relative Velocity
Vector subtraction gives the velocity of one object as observed from another moving object.
v_rel = v_A − v_B
📟 Try it — Relative Velocity Calculator
Enter both velocity vectors, then press Calculate…
🎮
Quick-Fire Vectors Game
12 questions on vector basics and products. How many can you get?
⚡ Vectors Challenge
Score: 0Question: 1/12Streak: 0🔥
Press Start to begin!
📝
Practice Questions
Graded from warm-up to challenge. Attempt fully before revealing the working.
Question 1 🥉2 marks
Find the magnitude of v = 5i + 12j.
√(25+144)=√169=13
Question 2 🥉2 marks
Add a=2i+3j and b=4i−1j.
6i + 2j
Question 3 🥉2 marks
Find a·b for a=2i+3j, b=4i+1j.
(2)(4)+(3)(1)=8+3=11
Question 4 🥉2 marks
State what quantity a·b directly gives when a is force and b is displacement.
Work done
Question 5 🥈4 marks
Find the direction (angle from positive x-axis) of v=7i+7j.
tanθ=7/7=1
θ=45°
θ=45°
Question 6 🥈4 marks
Find the resultant magnitude of a=9i+12j and b=−3i+4j added together.
Sum=6i+16j
Magnitude=√(36+256)=√292
≈17.09
Magnitude=√(36+256)=√292
≈17.09
Question 7 🥈3 marks
Find the 2D cross product a×b for a=2i+3j, b=4i+1j.
(2)(1)−(3)(4)=2−12=−10
Question 8 🥈4 marks
Two vectors a=1i+0j and b=0i+1j. Find a·b and state what this tells you about their relationship.
(1)(0)+(0)(1)=0 — a and b are orthogonal (perpendicular)
Question 9 🥇5 marks
Three forces act on a joint: F₁=(20,10)N, F₂=(−5,15)N, F₃=(10,−20)N. Find the resultant force's magnitude.
Sum x=20−5+10=25
Sum y=10+15−20=5
Magnitude=√(625+25)=√650
≈25.50N
Sum y=10+15−20=5
Magnitude=√(625+25)=√650
≈25.50N
Question 10 🥇5 marks
Find the angle between a=6i+8j and b=3i+4j using the dot product.
a·b=(6)(3)+(8)(4)=18+32=50
|a|=10, |b|=5
cosθ=50/(10×5)=1
θ=0° (parallel vectors)
|a|=10, |b|=5
cosθ=50/(10×5)=1
θ=0° (parallel vectors)
Question 11 🥇5 marks
A force F=(60,−20)N acts through displacement d=(3,5)m. Find the work done and comment on the sign.
W=(60)(3)+(−20)(5)=180−100=80 J — positive, meaning the force does net positive work in the direction of motion
Question 12 🥇5 marks
A wrench applies force F=(0,150)N at position r=(0.3,0)m from a bolt's pivot. Find the moment (torque) and state its rotational sense.
M=r×F=(0.3)(150)−(0)(0)=45 N·m — positive value indicates a counter-clockwise turning effect about the pivot