Vector Operations
Master vector addition, subtraction, scalar multiplication, and the dot product in two dimensions.
Vector Addition and Subtraction
Vectors are added and subtracted component by component. If a = (a1, a2) and b = (b1, b2):
a + b = (a1 + b1, a2 + b2)
a − b = (a1 − b1, a2 − b2)
Geometrically, addition follows the triangle rule or parallelogram rule: place the tail of b at the head of a, and the resultant goes from the tail of a to the head of b.
Scalar Multiplication
Multiplying a vector by a scalar (a real number) scales its magnitude while preserving (or reversing) its direction:
ka = k(a1, a2) = (ka1, ka2)
k > 0: Same direction, magnitude scaled by k.
k < 0: Opposite direction, magnitude scaled by |k|.
The Dot Product (Scalar Product)
The dot product of two vectors produces a scalar value. It has two equivalent definitions:
a · b = a1b1 + a2b2 (component form)
a · b = |a||b| cos θ (geometric form)
A key property: if a · b = 0 and neither vector is the zero vector, then a and b are perpendicular.
Key Vocabulary
Resultant Vector
The single vector obtained from adding two or more vectors together.
Scalar Multiplication
Scaling a vector by a real number, changing its magnitude but not (necessarily) its direction.
Dot Product
An operation on two vectors that produces a scalar, equal to a1b1 + a2b2.
Perpendicular
Two vectors are perpendicular (at 90°) when their dot product equals zero.
Worked Examples
If a = (2, 5) and b = (3, −1), find a + b and a − b.
Addition: a + b = (2 + 3, 5 + (−1)) = (5, 4).
Subtraction: a − b = (2 − 3, 5 − (−1)) = (−1, 6).
Find 3a where a = (4, −2).
Step 1: Multiply each component by 3: 3(4, −2) = (12, −6).
Answer: 3a = (12, −6). The magnitude is tripled, direction unchanged.
Find a · b and the angle between them, given a = (3, 4) and b = (4, −3).
Step 1: a · b = (3)(4) + (4)(−3) = 12 − 12 = 0.
Step 2: Since the dot product is 0, the vectors are perpendicular.
Answer: a · b = 0, and the angle between them is 90°.
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
If a = (1, 3) and b = (4, 2), what is a + b?
Question 2
What is −2 × (3, −4)?
Question 3
Find the dot product of (2, 7) and (3, −1).
Question 4
Two vectors have a dot product of 0. What can you conclude?
Question 5
If a = (1, 2) and b = (3, −5), what is 2a + b?
Key Concepts Summary
- ● Add or subtract vectors component by component.
- ● Scalar multiplication scales each component by the scalar value.
- ● The dot product gives a scalar: a · b = a1b1 + a2b2.
- ● If the dot product is zero, the vectors are perpendicular.
- ● The angle between vectors can be found using cos θ = (a · b) / (|a||b|).