Vectors, Lines & Planes Reference

Definitions

Magnitude

∥ v ∥

The length or size of a vector.

Example: The magnitude of

⟨ 3, 4 ⟩

5

Direction

The orientation in which a vector points.

Example: A direction angle locates a 2D vector from the positive

x

-axis.

Dot product

a \cdot b

A scalar product that measures directional alignment.

Example: Perpendicular nonzero vectors have dot product 0.

Cross product

a \times b

A 3D vector perpendicular to two input vectors.

Example: Its magnitude gives parallelogram area.

Projection

proj_{b} a

The vector component of one vector in the direction of another.

Example: Projection measures how much of

a

points along

b

Collinearity

Points or vectors lying on one line; direction vectors are scalar multiples.

Example:

⟨ 2, 4 ⟩

and

⟨ 1, 2 ⟩

are collinear.

Linear dependence

A set of vectors where at least one can be formed from the others.

Example: Two nonzero collinear vectors are dependent.

Basis

A linearly independent set that spans a vector space.

Example:

⟨ 1, 0 ⟩

and

⟨ 0, 1 ⟩

form a basis for

R^{2}

Line

r = r_{0} + t d

A set of points through one point in a fixed direction.

Example:

t

moves along the line.

Plane

n \cdot (r - r_{0}) = 0

A flat 2D set in 3D determined by a point and two directions, or a normal vector.

Example: The normal is perpendicular to every in-plane direction.

Formula sheet

10 formulas

Vector magnitude

∥ v ∥ = v_{1}^{2} + v_{2}^{2} + v_{3}^{2}

Variables: $v_{i}$ : vector components
Remember: Length is always nonnegative.

Dot product

a \cdot b = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}

Variables: $a_{i}, b_{i}$ : components
Remember: A scalar result.

Angle between vectors

cos θ = \frac{a \cdot b}{∥ a ∥∥ b ∥}

Variables: nonzero vectors
Remember: Dot product $0$ means perpendicular.

Cross product

a \times b = i a_{1} b_{1} j a_{2} b_{2} k a_{3} b_{3}

Variables: 3D vectors
Remember: Result is perpendicular to both.

Projection

proj_{b} a = \frac{a \cdot b}{∥ b ∥ ^{2}} b

Variables: $b \neq = 0$
Remember: Vector component of $a$ along $b$ .

Direction cosines

cos α = \frac{a _{1}}{∥ a ∥}, cos β = \frac{a _{2}}{∥ a ∥}, cos γ = \frac{a _{3}}{∥ a ∥}

Variables: $α, β, γ$ : angles with positive axes
Remember: Squares of the direction cosines sum to $1$ .

Vector line

r = r_{0} + t d

Variables: $r_{0}$ : point vector; $d$ : direction; $t \in R$
Remember: One parameter moves along a line.

Scalar plane

A x + B y + C z + D = 0

Variables: $⟨ A, B, C ⟩$ : normal vector
Remember: The coefficients form a normal.

Plane from point and normal

n \cdot (r - r_{0}) = 0

Variables: $n$ : normal; $r_{0}$ : known point
Remember: Every in-plane displacement is perpendicular to $n$ .

Parallelogram and triangle area

A_{p a r a} = ∥ a \times b ∥, A_{t r i} = \frac{1}{2} ∥ a \times b ∥

Variables: $a, b$ : adjacent side vectors
Remember: Cross-product magnitude measures area.

Worked examples

Find vector magnitude

Find the magnitude of $v = ⟨ 2, - 3, 6 ⟩$ .

Show solution

Square the components: $2^{2} + (- 3)^{2} + 6^{2}$ .
Add: $4 + 9 + 36 = 49$ .
Take the nonnegative square root: $∥ v ∥ = 49 = 7$ .

Comparison table

Line equations in 2D/3D
Form	Structure	Best use
Vector	$r = r_{0} + t d$	Compact point-and-direction representation
Parametric	$x = x_{0} + a t, y = y_{0} + b t, z = z_{0} + c t$	Coordinates as functions of one parameter
Scalar (2D)	$A x + B y + C = 0$	Normal-based line relationships in a plane
Symmetric	$\frac{x - x _{0}}{a} = \frac{y - y _{0}}{b} = \frac{z - z _{0}}{c}$	Eliminates the parameter when components are nonzero

Variable glossary

$r$: position vector
$r_{0}$: known point vector
$d$: direction vector
$n$: normal vector
$t, s$: real parameters

Student tools

Open graph studio Guided calculators