Vectors
Grade 12 · Math · Study Guide
Understand vector quantities, operations, and their geometric meaning in two and three dimensions.
A vector has both magnitude and direction, unlike a scalar which has magnitude only. Vectors in 2D are written as <a, b> or ai + bj; in 3D as <a, b, c> or ai + bj + ck. The magnitude is ||v|| = sqrt(a²+b²) in 2D.
Key operations include addition (add components), scalar multiplication (scale each component), and the dot product: u·v = u₁v₁ + u₂v₂ + u₃v₃. The dot product equals ||u||·||v||·cos θ, making it useful for finding angles between vectors.
The cross product u × v only applies in 3D and produces a new vector perpendicular to both u and v. Its magnitude equals ||u||·||v||·sin θ, which equals the area of the parallelogram formed by u and v.
| Operation | Formula |
|---|---|
| Addition | <a,b> + <c,d> = <a+c, b+d> |
| Scalar mult. | k<a,b> = <ka, kb> |
| Magnitude | ||<a,b>|| = sqrt(a²+b²) |
| Dot product | <a,b>·<c,d> = ac + bd |
| Unit vector | û = v / ||v|| |
| Result | Meaning |
|---|---|
| u·v = 0 | Vectors are perpendicular |
| u·v > 0 | Angle between them < 90° |
| u·v < 0 | Angle between them > 90° |
Key terms
- Vector
- A quantity with both magnitude and direction.
- Dot product
- u·v = ||u||·||v||·cos θ; a scalar measuring directional alignment.
- Cross product
- u × v — a 3D vector perpendicular to both u and v with magnitude ||u||·||v||·sin θ.
- Unit vector
- A vector with magnitude 1; found by dividing a vector by its magnitude.
Practice questions
What is the magnitude of v = <3, 4>?
- 5
- 7
- 12
- 25
Show answer
||v|| = sqrt(3²+4²) = sqrt(9+16) = sqrt(25) = 5.
Find the dot product of u = <2, -1, 3> and v = <4, 5, -2>.
Show answer
u·v = 2(4) + (-1)(5) + 3(-2) = 8 - 5 - 6 = -3.
If u·v = 0, what is the relationship between u and v?
- Parallel
- Perpendicular
- Equal
- Antiparallel
Show answer
A dot product of 0 means cos θ = 0, so θ = 90° — the vectors are perpendicular.
Find the unit vector in the direction of v = <6, 8>.
Show answer
||v|| = sqrt(36+64) = 10. û = <6/10, 8/10> = <3/5, 4/5>.
The cross product of two parallel vectors is the zero vector.
Show answer
Parallel vectors have θ = 0 or π, so sin θ = 0 and ||u × v|| = 0.
Fun facts
- Dot products power machine learning — neural network activations are computed as dot products of weights and inputs.
- The cross product is anticommutative: u × v = −(v × u), so order matters unlike multiplication of numbers.
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