Vectors in Two and Three Dimensions

Introduction: Why Vectors Matter 🌍

students, engineers use vectors to describe both size and direction. A vector can represent a force pushing a bridge, the velocity of a drone, or the displacement of a moving robot. Unlike a simple number, a vector tells us not only how much but also which way.

In this lesson, you will learn how vectors work in two and three dimensions, how to write them in component form, and how to use them in geometric and trigonometric problems. These ideas connect directly to the wider study of trigonometry and geometry because vectors rely on coordinates, angles, and distance. By the end, you should be able to explain vector terminology, carry out basic vector calculations, and use vectors to solve real-world problems.

What Is a Vector?

A vector is a quantity that has both magnitude and direction. Magnitude means the length or size of the vector, while direction tells us where it points. A quantity such as mass is a scalar because it has size only. A quantity such as displacement is a vector because moving $5\,\text{m}$ north is different from moving $5\,\text{m}$ east.

A vector is often shown as an arrow. The arrow’s length represents its magnitude, and the arrow’s orientation represents its direction. In writing, vectors are commonly shown in bold, such as $\mathbf{v}$, or with an arrow, such as $\vec{v}$.

For example, if a wind blows with velocity $\mathbf{v}$, then $\mathbf{v}$ may describe both speed and direction. In engineering, this is useful because many physical quantities depend on direction, not just size.

Vectors in Two Dimensions

In two dimensions, vectors lie on a flat plane. We usually use the $x$-axis and $y$-axis to describe them. A vector in component form is written as

$$\mathbf{v} = \begin{pmatrix} a \\ b \end{pmatrix}$$

where $a$ is the horizontal component and $b$ is the vertical component.

The vector from the origin to the point $(a,b)$ is the position vector of that point. If a vector moves $3$ units right and $4$ units up, then its component form is

$$\begin{pmatrix} 3 \\ 4 \end{pmatrix}$$

Its magnitude is found using Pythagoras’ theorem:

$$|\mathbf{v}| = \sqrt{a^2 + b^2}$$

So for $\mathbf{v} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$,

$$|\mathbf{v}| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$$

This is a classic example of geometry and trigonometry working together. The vector forms a right triangle with the axes, and the magnitude is the hypotenuse.

Components, Direction, and Unit Vectors

A very useful idea is the unit vector, which has magnitude $1$. In two dimensions, the standard unit vectors are

$$\mathbf{i} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad \mathbf{j} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$$

Any vector $\mathbf{v} = \begin{pmatrix} a \\ b \end{pmatrix}$ can be written as

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

This form is helpful because it shows the contribution of each axis clearly.

If a vector has magnitude $r$ and makes an angle $\theta$ with the positive $x$-axis, then its components are

$$a = r\cos\theta, \quad b = r\sin\theta$$

So the vector can be written as

$$\mathbf{v} = r\cos\theta\,\mathbf{i} + r\sin\theta\,\mathbf{j}$$

This is where trigonometry enters the picture. The cosine and sine functions convert between polar-style information $(r,\theta)$ and rectangular components $(a,b)$.

Example

Suppose a force has magnitude $10$ units and acts at an angle of $30^\circ$ above the positive $x$-axis. Then

$$a = 10\cos 30^\circ = 10\cdot \frac{\sqrt{3}}{2} = 5\sqrt{3}$$

$$b = 10\sin 30^\circ = 10\cdot \frac{1}{2} = 5$$

So the vector is

$$\mathbf{v} = \begin{pmatrix} 5\sqrt{3} \\ 5 \end{pmatrix}$$

This kind of decomposition is used in engineering to break one force into horizontal and vertical parts.

Vector Operations

Vectors can be added, subtracted, and multiplied by scalars.

$$\mathbf{u} = \begin{pmatrix} u_1 \\ u_2 \end{pmatrix}, \quad \mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix}$$

then

$$\mathbf{u} + \mathbf{v} = \begin{pmatrix} u_1 + v_1 \\ u_2 + v_2 \end{pmatrix}$$

and

$$\mathbf{u} - \mathbf{v} = \begin{pmatrix} u_1 - v_1 \\ u_2 - v_2 \end{pmatrix}$$

Scalar multiplication changes the length of a vector, and if the scalar is negative, it also reverses the direction.

$$k\mathbf{v} = \begin{pmatrix} ka \\ kb \end{pmatrix}$$

Example

Let

$$\mathbf{u} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}, \quad \mathbf{v} = \begin{pmatrix} -3 \\ 4 \end{pmatrix}$$

Then

$$\mathbf{u} + \mathbf{v} = \begin{pmatrix} -1 \\ 3 \end{pmatrix}$$

and

$$2\mathbf{u} - \mathbf{v} = \begin{pmatrix} 4 \\ -2 \end{pmatrix} - \begin{pmatrix} -3 \\ 4 \end{pmatrix} = \begin{pmatrix} 7 \\ -6 \end{pmatrix}$$

These operations are useful when combining displacements or balancing forces.

Vectors in Three Dimensions

In three dimensions, vectors also include a $z$-component. A vector is written as

$$\mathbf{v} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}$$

The standard unit vectors are

$$\mathbf{i} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \quad \mathbf{j} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \quad \mathbf{k} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$$

So the vector can also be written as

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

The magnitude in three dimensions is

$$|\mathbf{v}| = \sqrt{a^2 + b^2 + c^2}$$

Example

$$\mathbf{v} = \begin{pmatrix} 2 \\ 3 \\ 6 \end{pmatrix}$$

then its magnitude is

$$|\mathbf{v}| = \sqrt{2^2 + 3^2 + 6^2} = \sqrt{49} = 7$$

Three-dimensional vectors are important in fields such as robotics, navigation, and structural analysis, where motion and force occur in space rather than on a flat surface.

Direction, Distance, and Geometry

Vectors help us describe the line segment between two points. If points $A(x_1,y_1)$ and $B(x_2,y_2)$ are in the plane, then the vector from $A$ to $B$ is

$$\overrightarrow{AB} = \begin{pmatrix} x_2 - x_1 \\ y_2 - y_1 \end{pmatrix}$$

In three dimensions, if $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$, then

$$\overrightarrow{AB} = \begin{pmatrix} x_2 - x_1 \\ y_2 - y_1 \\ z_2 - z_1 \end{pmatrix}$$

The distance between the points is the magnitude of this vector.

Example

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

$$\overrightarrow{AB} = \begin{pmatrix} 4 \\ 3 \end{pmatrix}$$

so the distance is

$$\sqrt{4^2 + 3^2} = 5$$

This is another use of the Pythagorean theorem in coordinate geometry.

The Dot Product and Angles

The dot product is a way to multiply two vectors and get a scalar. In two dimensions,

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

In three dimensions,

$$\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3$$

The dot product is very useful for finding the angle between vectors. If $\theta$ is the angle between nonzero vectors $\mathbf{u}$ and $\mathbf{v}$, then

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

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

Example

Let

$$\mathbf{u} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}, \quad \mathbf{v} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}$$

Then

$$\mathbf{u} \cdot \mathbf{v} = 1\cdot 2 + 2\cdot 1 = 4$$

Also,

$$|\mathbf{u}| = \sqrt{1^2 + 2^2} = \sqrt{5}, \quad |\mathbf{v}| = \sqrt{2^2 + 1^2} = \sqrt{5}$$

$$\cos\theta = \frac{4}{5}$$

This shows how vectors and trigonometry combine to measure angles in a precise way.

Real-World Engineering Meaning ⚙️

Vectors are not just abstract math. They are widely used in engineering. A cable tension force has a direction along the cable. A ship’s movement is affected by current and wind, each of which can be modeled as vectors. In computer graphics, position and motion in $3$D space are handled using vectors. In civil engineering, loads on beams and bridges are often split into vector components to understand how structures respond.

For example, if a drone experiences a sideways wind force and a forward thrust force, the total effect can be found by adding the force vectors. This makes it possible to predict the drone’s path.

Conclusion

Vectors in two and three dimensions give a powerful way to describe direction and magnitude together. They connect directly to geometry through coordinates and distances, and to trigonometry through sine, cosine, and angle relationships. For students, understanding vectors is an important step in Engineering Mathematics because many real systems in physics and engineering depend on them. Once you can represent vectors, find their magnitudes, add them, and use the dot product, you have a strong foundation for more advanced topics.

Study Notes

A vector has both magnitude and direction.
Scalars have magnitude only; vectors have magnitude and direction.
In two dimensions, a vector is written as $\begin{pmatrix} a \\ b \end{pmatrix}$.
In three dimensions, a vector is written as $\begin{pmatrix} a \\ b \\ c \end{pmatrix}$.
The magnitude of $\begin{pmatrix} a \\ b \end{pmatrix}$ is $\sqrt{a^2 + b^2}$.
The magnitude of $\begin{pmatrix} a \\ b \\ c \end{pmatrix}$ is $\sqrt{a^2 + b^2 + c^2}$.
Standard unit vectors are $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$.
Any vector can be written using unit vectors, such as $a\mathbf{i} + b\mathbf{j} + c\mathbf{k}$.
Vector addition and subtraction are done component by component.
Scalar multiplication changes the size of a vector and may reverse its direction.
Trigonometry helps convert between magnitude-angle form and component form.
The dot product is used to find angles between vectors and is a scalar.
Vectors are essential in engineering applications such as force, motion, navigation, and structural analysis.