Vector Operations
Addition, Subtraction, and Scalar Multiplication
SPEC 10.3 • VECTORS
Why Vector Operations?
Vector operations let us combine and manipulate vectors algebraically, just like we do with numbers.
Real-World Uses
Adding forces on an object
Combining velocities
Finding resultant displacements
Navigation and motion
What We'll Cover
Vector addition
Vector subtraction
Scalar multiplication
Combined operations
Key Idea
Vector operations work component by component - operate on each coordinate separately, then combine the results.
Vector Addition - 2D
To add vectors, add the corresponding components:
Formula
ab + cd = a + cb + d
32 + 14 = 3 + 12 + 4 = 46
5−3 + −27 = 5 + (−2)−3 + 7 = 34
Remember
Add x-components together, add y-components together. Keep them separate!
Vector Addition - 3D
The same principle extends to three dimensions:
Formula
abc + def = a + db + ec + f
23−1 + 4−25 = 2 + 43 + (−2)−1 + 5 = 614
i-j-k Notation
(2i + 3j − k) + (4i − 2j + 5k) = 6i + j + 4k
Collect like terms: i's with i's, j's with j's, k's with k's.
Properties of Vector Addition
PropertyStatementMeaning
Commutativea + b = b + aOrder doesn't matter
Associative( a + b ) + c = a + ( b + c )Grouping doesn't matter
Identitya + 0 = aAdding zero gives same vector
Inversea + (− a ) = 0Vector plus its negative = zero
Exam Tip
These properties let you rearrange and group vectors however is most convenient for calculation.
Worked Example01
Given a = 4−2 and b = 37, find a + b.
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1. Add the x-components: 4 + 3 = 7
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2. Add the y-components: −2 + 7 = 5
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3. Write the result: a + b = 75
Triangle Rule for Addition
Geometrically, to add vectors, place them "nose to tail":
Triangle Rule
Place the tail of b at the head of a. The resultant a + b goes from the start of a to the end of b.
Parallelogram Rule for Addition
If both vectors start at the same point, complete the parallelogram:
Order Doesn't Matter
Whether you go a then b (solid → dashed blue) or b then a (solid → dashed green), you reach the same endpoint. This shows a + b = b + a geometrically.
Vector Subtraction
To subtract vectors, subtract the corresponding components:
Formula
ab − cd = a − cb − d
58 − 23 = 5 − 28 − 3 = 35
4−1 − 7−6 = 4 − 7−1 − (−6) = −35
Watch Out
Be careful with signs! In the second example: −1 − (−6) = −1 + 6 = 5
Worked Example02
Given p = 3i + 2j − 4k and q = i − 5j + 2k, find p + q.
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1. Collect i terms: 3 + 1 = 4
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2. Collect j terms: 2 + (−5) = −3
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3. Collect k terms: −4 + 2 = −2
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4.p + q = 4i − 3j − 2k
Worked Example03
Given a = 6−2 and b = 45, find a − b.
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1. Subtract x-components: 6 − 4 = 2
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2. Subtract y-components: −2 − 5 = −7
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3.a − b = 2−7
Scalar Multiplication
To multiply a vector by a scalar (number), multiply each component:
Formula
kab = kakb kabc = kakbkc
32−4 = 3 × 23 × (−4) = 6−12
−251−3 = −10−26
Effect on Vector
k > 0: Same direction, magnitude multiplied by k
k < 0: Opposite direction, magnitude multiplied by |k|
Worked Example04
Given v = −325, find: (a) 4v (b) −½v
(a) 4v:
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Multiply each component by 4: 4−325 = −12820
(b) −½v:
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Multiply each component by −½: −½−325 = 3/2−1−5/2
Parallel Vectors
Definition
Two vectors a and b are parallel if one is a scalar multiple of the other:b = ka for some scalar k ≠ 0
Same Direction (k > 0)
69 = 323
These point the same way
Opposite Direction (k < 0)
−4−6 = −223
These point opposite ways
How to Check
Check if the ratio of corresponding components is the same: if b₁/a₁ = b₂/a₂ = b₃/a₃ = k, they're parallel.
Worked Example05
Given a = 2−1 and b = 34, find 2a + 3b.
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1. Find 2a: 22−1 = 4−2
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2. Find 3b: 334 = 912
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3. Add: 4−2 + 912 = 4 + 9−2 + 12 = 1310
Worked Example06
Vectors a = 4−6 and b = −10k are parallel. Find the value of k.
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1. For parallel vectors: b = λa
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2. From x-components: −10 = λ(4), so λ = −2.5
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3. From y-components: k = λ(−6) = (−2.5)(−6) = 15
k = 15
Worked Example07
Given a = 12−1, b = 302, c = −113, find a − 2b + 3c.
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1. 2b = 604, 3c = −339
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2. x: 1 − 6 + (−3) = −8
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3. y: 2 − 0 + 3 = 5
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4. z: −1 − 4 + 9 = 4
a − 2b + 3c = −854
Worked Example08
Given a = 23 and b = 1−2, find the value of k such that a + kb = 5−3.
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1.a + kb = 23 + k1−2 = 2+k3−2k
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2. From x-component: 2 + k = 5, so k = 3
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3. Check with y-component: 3 − 2(3) = 3 − 6 = −3 ✓
k = 3
Angles with Axes in 3D
A 3D vector makes angles with each coordinate axis. These are called direction angles.
Direction Cosines
For vector v = abc with magnitude |v|: cos α = a/|v| cos β = b/|v| cos γ = c/|v| where α is angle with x-axis, β with y-axis, γ with z-axis
To Find an Angleα = cos⁻¹(a/|v|)
Use the relevant component divided by the magnitude
Key Propertycos²α + cos²β + cos²γ = 1
Direction cosines always satisfy this identity!
Worked Example09
Find the angles that the vector v = 221 makes with each of the coordinate axes.
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1. Find magnitude: |v| = √(2² + 2² + 1²) = √(4 + 4 + 1) = √9 = 3
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2. Angle with x-axis: cos α = 2/3, so α = cos⁻¹(2/3) = 48.2°
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3. Angle with y-axis: cos β = 2/3, so β = cos⁻¹(2/3) = 48.2°
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4. Angle with z-axis: cos γ = 1/3, so γ = cos⁻¹(1/3) = 70.5°
α = 48.2°, β = 48.2°, γ = 70.5°
Practice
Question 1: Given p = 2−34 and q = −152, find p + q.
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x: 2 + (−1) = 1
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y: −3 + 5 = 2
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z: 4 + 2 = 6
p + q = 126
Practice
Question 2: Given a = 4−1 and b = 23, find 3a − 2b.
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3a = 12−3
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2b = 46
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3a − 2b = 12−4−3−6 = 8−9
Practice
Question 3: Given a = 14 and b = 2−1, find the value of k such that a + kb = 71.
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a + kb = 1+2k4−k = 71
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From x: 1 + 2k = 7, so 2k = 6, k = 3
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Check y: 4 − 3 = 1 ✓
k = 3
Summary
OperationRuleExample
AdditionAdd corresponding components2 3 + 1 4 = 3 7
SubtractionSubtract corresponding components5 2 − 3 4 = 2 −2
Scalar ×Multiply each component by scalar3 2 −1 = 6 −3
Parallelb = k a for some k ≠ 06 9 = 3 2 3
Key Exam Skills
Work component by component - keep x, y, z separate
Apply scalar multiplication before adding/subtracting
Watch signs carefully, especially with subtraction
For parallel vectors, find and use the scalar multiplier
Next: Why Vector Operations?