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Vectors

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1. What a vector is

A vector is an object with both magnitude and direction. In geometry and physics, vectors represent quantities such as displacement, velocity, acceleration, and force.

In algebra, vectors are often treated as ordered lists of numbers in $\mathbb{R}^n$:

$$ \mathbf{v} = \langle v_1, v_2, \dots, v_n \rangle $$

The numbers $v_1, v_2, \dots, v_n$ are the components of the vector.

Geometric view

In 2D or 3D, a vector can be drawn as an arrow:

  • Length represents magnitude

  • Arrow direction represents orientation

  • The vector is independent of its position in space

Two vectors are equal if they have the same magnitude and direction, even if they start at different points.

Free vectors and position vectors

  • A free vector can be translated anywhere without changing it.

  • A position vector starts at the origin and ends at a point.

If $P = (x, y, z)$, then the position vector of $P$ is

$$ \mathbf{p} = \langle x, y, z \rangle $$

2. Notation and components

Vectors may be written in several equivalent forms:

$$ \mathbf{v} = \langle 3, -2 \rangle $$
$$ \mathbf{v} = \begin{bmatrix} 3 \\ -2 \end{bmatrix} $$
$$ \mathbf{v} = 3\mathbf{i} - 2\mathbf{j} $$

In 3D:

$$ \mathbf{v} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k} $$

where $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ are the standard basis vectors:

$$ \mathbf{i} = \langle 1, 0, 0 \rangle,\quad \mathbf{j} = \langle 0, 1, 0 \rangle,\quad \mathbf{k} = \langle 0, 0, 1 \rangle $$

Component form from two points

If a vector goes from point $A(x_1, y_1)$ to point $B(x_2, y_2)$, then

$$ \overrightarrow{AB} = \langle x_2 - x_1,\ y_2 - y_1 \rangle $$

In 3D:

$$ \overrightarrow{AB} = \langle x_2 - x_1,\ y_2 - y_1,\ z_2 - z_1 \rangle $$

This is one of the most common ways vectors appear in geometry problems.

3. Magnitude and unit vectors

The magnitude or length of $\mathbf{v} = \langle v_1, v_2, \dots, v_n \rangle$ is

$$ \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2} $$

In 2D:

$$ \|\mathbf{v}\| = \sqrt{x^2 + y^2} $$

In 3D:

$$ \|\mathbf{v}\| = \sqrt{x^2 + y^2 + z^2} $$

Unit vectors

A unit vector has length 1. The unit vector in the direction of $\mathbf{v}$ is

$$ \hat{\mathbf{v}} = \frac{\mathbf{v}}{\|\mathbf{v}\|} $$

provided $\mathbf{v} \neq \mathbf{0}$.

Example:

If $\mathbf{v} = \langle 3, 4 \rangle$, then

$$ \|\mathbf{v}\| = 5,\qquad \hat{\mathbf{v}} = \left\langle \frac{3}{5}, \frac{4}{5} \right\rangle $$

Direction in 2D

If $\mathbf{v} = \langle x, y \rangle$, then the direction angle $\theta$ satisfies

$$ \tan \theta = \frac{y}{x} $$

with the correct quadrant chosen from the signs of $x$ and $y$.

4. Basic vector operations

Addition and subtraction

For $\mathbf{u} = \langle u_1, u_2, \dots \rangle$ and $\mathbf{v} = \langle v_1, v_2, \dots \rangle$:

$$ \mathbf{u} + \mathbf{v} = \langle u_1 + v_1,\ u_2 + v_2,\ \dots \rangle $$
$$ \mathbf{u} - \mathbf{v} = \langle u_1 - v_1,\ u_2 - v_2,\ \dots \rangle $$

Geometrically, addition follows the tip-to-tail or parallelogram rule.

Scalar multiplication

For scalar $c$:

$$ c\mathbf{v} = \langle cv_1, cv_2, \dots, cv_n \rangle $$

Effects:

  • If $|c| > 1$, the vector stretches

  • If $0 < |c| < 1$, the vector shrinks

  • If $c < 0$, the direction reverses

Zero vector

The zero vector is

$$ \mathbf{0} = \langle 0, 0, \dots, 0 \rangle $$

It has zero magnitude and no direction.

Interactive visual

Vector addition

Change the x and y components of two vectors and watch the resultant update in real time.

Resultant (3, 7)
Magnitude 7.62

5. Dot product

The dot product of two vectors in $\mathbb{R}^n$ is

$$ \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + \cdots + u_nv_n $$

In 2D:

$$ \langle a, b \rangle \cdot \langle c, d \rangle = ac + bd $$

In 3D:

$$ \langle a, b, c \rangle \cdot \langle d, e, f \rangle = ad + be + cf $$

Angle between vectors

The dot product also satisfies

$$ \mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \, \|\mathbf{v}\| \cos \theta $$

where $\theta$ is the angle between the vectors.

So

$$ \cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \, \|\mathbf{v}\|} $$

This is often the fastest way to find an angle.

Orthogonality

Two nonzero vectors are orthogonal if their dot product is zero:

$$ \mathbf{u} \cdot \mathbf{v} = 0 $$

This means the vectors are perpendicular.

Projection idea from the dot product

The scalar component of $\mathbf{v}$ along $\mathbf{u}$ is

$$ \operatorname{comp}_{\mathbf{u}} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{u}}{\|\mathbf{u}\|} $$

The vector projection is covered in the next section.

6. Cross product

The cross product is defined for vectors in $\mathbb{R}^3$.

For

$$ \mathbf{u} = \langle u_1, u_2, u_3 \rangle,\qquad \mathbf{v} = \langle v_1, v_2, v_3 \rangle $$

the cross product is

$$ \mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix} $$

which expands to

$$ \mathbf{u} \times \mathbf{v} = \langle u_2v_3 - u_3v_2,\ u_3v_1 - u_1v_3,\ u_1v_2 - u_2v_1 \rangle $$

Geometric meaning

  • $\mathbf{u} \times \mathbf{v}$ is perpendicular to both $\mathbf{u}$ and $\mathbf{v}$

  • Its magnitude is

$$ \|\mathbf{u} \times \mathbf{v}\| = \|\mathbf{u}\| \, \|\mathbf{v}\| \sin \theta $$

  • It gives the area of the parallelogram spanned by the two vectors

So the triangle area is

$$ \frac{1}{2}\|\mathbf{u} \times \mathbf{v}\| $$

Orientation

The cross product is not commutative:

$$ \mathbf{u} \times \mathbf{v} = -(\mathbf{v} \times \mathbf{u}) $$

Use the right-hand rule to determine direction.

7. Vector projections

The projection of $\mathbf{v}$ onto $\mathbf{u}$ is the component of $\mathbf{v}$ that points along $\mathbf{u}$.

Scalar projection

$$ \operatorname{comp}_{\mathbf{u}} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{u}}{\|\mathbf{u}\|} $$

Vector projection

$$ \operatorname{proj}_{\mathbf{u}} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u} $$

This formula is useful in decomposition problems and least-squares geometry.

Orthogonal decomposition

Any vector can be split into parallel and perpendicular parts relative to $\mathbf{u}$:

$$ \mathbf{v} = \operatorname{proj}_{\mathbf{u}} \mathbf{v} + \mathbf{v}_\perp $$

where

$$ \mathbf{v}_\perp = \mathbf{v} - \operatorname{proj}_{\mathbf{u}} \mathbf{v} $$

and $\mathbf{v}_\perp \cdot \mathbf{u} = 0$.

8. Lines and planes

Vectors provide compact equations for geometric objects.

Line in 2D or 3D

A line through point $\mathbf{r}_0$ with direction vector $\mathbf{d}$ has vector equation

$$ \mathbf{r}(t) = \mathbf{r}_0 + t\mathbf{d} $$

where $t$ is a parameter.

In components:

$$ x = x_0 + at,\qquad y = y_0 + bt,\qquad z = z_0 + ct $$

if $\mathbf{d} = \langle a, b, c \rangle$.

Plane in 3D

A plane through point $\mathbf{r}_0$ with normal vector $\mathbf{n}$ satisfies

$$ \mathbf{n} \cdot (\mathbf{r} - \mathbf{r}_0) = 0 $$

If $\mathbf{n} = \langle A, B, C \rangle$ and $\mathbf{r}_0 = (x_0, y_0, z_0)$, then

$$ A(x - x_0) + B(y - y_0) + C(z - z_0) = 0 $$

This is the standard point-normal form of a plane.

Distance from a point to a plane

For plane

$$ Ax + By + Cz + D = 0 $$

and point $(x_0, y_0, z_0)$, the distance is

$$ \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}} $$

9. Common workflows

Find the vector from one point to another

  1. Identify the initial point $A$ and terminal point $B$

  2. Subtract coordinates: $B - A$

  3. Simplify the components

Example:

If $A = (1, -3)$ and $B = (5, 2)$, then

$$ \overrightarrow{AB} = \langle 4, 5 \rangle $$

Find an angle between vectors

  1. Compute $\mathbf{u} \cdot \mathbf{v}$

  2. Compute $\|\mathbf{u}\|$ and $\|\mathbf{v}\|$

  3. Use

$$ \cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \, \|\mathbf{v}\|} $$

  1. Apply inverse cosine

Test perpendicularity or parallelism

  • Perpendicular: $\mathbf{u} \cdot \mathbf{v} = 0$

  • Parallel: one vector is a scalar multiple of the other

Build a line or plane

  • Line: point plus direction vector

  • Plane: point plus normal vector

If the problem gives a geometric picture, convert it to vectors first. The algebra is usually simpler afterward.

10. Formula summary

Core formulas

$$ \mathbf{u} + \mathbf{v} = \langle u_1 + v_1, \dots, u_n + v_n \rangle $$
$$ c\mathbf{v} = \langle cv_1, \dots, cv_n \rangle $$
$$ \|\mathbf{v}\| = \sqrt{v_1^2 + \cdots + v_n^2} $$
$$ \mathbf{u} \cdot \mathbf{v} = \sum_{i=1}^n u_i v_i $$
$$ \mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \, \|\mathbf{v}\| \cos \theta $$
$$ \mathbf{u} \times \mathbf{v} = \langle u_2v_3 - u_3v_2,\ u_3v_1 - u_1v_3,\ u_1v_2 - u_2v_1 \rangle $$
$$ \operatorname{proj}_{\mathbf{u}} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u} $$
$$ \mathbf{r}(t) = \mathbf{r}_0 + t\mathbf{d} $$
$$ \mathbf{n} \cdot (\mathbf{r} - \mathbf{r}_0) = 0 $$

11. Pitfalls

Sign mistakes in components

When subtracting coordinates, always use

$$ \overrightarrow{AB} = B - A $$

not the other way around unless the direction is intentionally reversed.

Confusing dot product with multiplication

The dot product produces a scalar, not a vector.

Confusing dot and cross products

  • Dot product measures alignment and angle

  • Cross product measures perpendicularity and area in 3D

Forgetting absolute value in distance formulas

Distance is always nonnegative. Use absolute value when needed.

Using the wrong dimension

  • Dot product works in any dimension

  • Cross product, in the standard form, is a 3D operation

Losing the direction of a vector

A vector is not just its length. Reversing a vector changes its direction even if the magnitude stays the same.

Final checks

Before moving on in a vector problem, verify:

  • The components point the correct way

  • The dimensions match the operation

  • Units are consistent if the vectors are physical quantities

  • The final answer has the right geometric meaning

Sources