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Vectors in Math — Definition, Types, and Worked Examples

TL;DR

A vector is a quantity with both magnitude and direction, drawn as an arrow whose length is the size and whose arrowhead is the direction. This article covers the types of vectors, how to write them in component form, the core operations (addition, scalar multiples, dot and cross products), and the errors students hit most.

What Is A Vector?

A vector is a quantity that has both a magnitude (a size) and a direction. You write it as a directed line segment — an arrow — where the length of the arrow shows the magnitude and the arrowhead shows the direction. The starting point is the tail and the ending point is the head.

Compare that with a scalar, a quantity that has size only. "50 km/h" is a scalar. "50 km/h heading west" is a vector, because the direction now matters. Displacement, velocity, acceleration, and force are all vectors; mass, temperature, and speed are scalars.

A vector is written several ways: with bold type, a\mathbf{a}, or with an overhead arrow, a⃗\vec{a}, or by naming its endpoints, AB⃗\vec{AB} (tail AAA, head BBB). In this article, we use the arrow notation a⃗\vec{a} throughout.

How Do You Write A Vector In Component Form?

In a coordinate plane, a vector from the origin to the point (x,y) is written in component form as a⃗=⟨x,y⟩\vec{a} = \langle x, y \rangle, or with unit vectors as a⃗=x,i^+y,j^\vec{a} = x,\hat{i} + y,\hat{j}. Here i^\hat{i} points one unit along the xxx-axis and j^\hat{j} points one unit along the yyy-axis. In three dimensions you add k^\hat{k} for the z-direction.

The magnitude of a⃗=x,i^+y,j^+z,k^\vec{a} = x,\hat{i} + y,\hat{j} + z,\hat{k} is given by:

∣a⃗∣=\sqrt{x^2 + y^2 + z^2}.

For a⃗=3,i^+4,j^, the magnitude is 5.

Variable glossary. x,y,z are the components; i^,j^,k^ are the unit vectors along the axes; ∣a⃗∣ is the magnitude.

What Are The Types Of Vectors?

• Zero vector (0⃗): magnitude 0, no defined direction.
• Unit vector (a^): magnitude exactly 1.
• Position vector: a vector from the origin to a point.
• Equal vectors: same magnitude and same direction.
• Negative vector: same magnitude, opposite direction.
• Parallel vectors: same or opposite direction.
• Collinear vectors: lie along the same line.
• Orthogonal vectors: perpendicular — the angle between them is 90°.
• Coplanar vectors: lying in one plane.

Examples of Vectors

Example 1

Find the magnitude of the vector a⃗=6,i^+8,j^.
∣a⃗∣=\sqrt{6^2 + 8^2} = 10.
Final answer: ∣a⃗∣=10.

Example 2

Find the unit vector in the direction of a⃗=3,i^−4,j^.
A student divides each component by the number of components: Wrong attempt:

a^=\frac{3}{2},\hat{i} - \frac{4}{2},\hat{j}.
Final answer: a^=\frac{3}{5},\hat{i} - \frac{4}{5},\hat{j}.

Example 3

Add a⃗=4,i^+2,j^−5,k^ and b⃗=3,i^−2,j^+k^.
Add matching components.

a⃗+b⃗=7,i^+0,j^−4,k^
Final answer: a⃗+b⃗=7,i^−4,k^.

Example 4

Find the dot product of a⃗=2,i^+3,j^ and b⃗=4,i^−j^.

a⃗⋅b⃗=5 Final answer: a⃗⋅b⃗=5; not orthogonal.

Example 5

Find the angle between a⃗=i^+j^ and b⃗=i^−j^.

a⃗⋅b⃗=0 Final answer: θ=90° — the vectors are orthogonal.

Example 6

A boat heads due north at 5 km/h while a current pushes it due east at 12 km/h.
∣R⃗∣=13
Final answer: the boat moves at 13 km/h.

Why Vectors Exist: "You Can't Navigate With A Number Alone"

Numbers alone describe how much. They cannot describe which way.
Three settings show why the idea earns its keep:

• Navigation and physics.
• Computer graphics.
• Engineering structures.

The operations come from one of two products. The dot product measures how much two vectors point the same way. The cross product builds a new vector perpendicular to both.

What Are The Most Common Mistakes With Vectors?

Mistake 1: Treating a vector as a plain number

What's the correct way? add component by component.

Mistake 2: Dividing by the component count to get a unit vector

What's the correct way? divide the whole vector by its magnitude.

Mistake 3: Mixing up the dot product and the cross product

What's the correct way?

Conclusion

• A vector carries both magnitude and direction; a scalar carries magnitude only.
• In component form a⃗=x,i^+y,j^+z,k^
• The main types: zero, unit, equal, negative, parallel, collinear, orthogonal, coplanar.
• The dot product returns a scalar; the cross product returns a perpendicular vector.

Frequently Asked Questions

Is a vector the same as a scalar?
No. A scalar has size only; a vector has size and direction.

How do you find the magnitude of a vector?
Use the formula ∣a⃗∣=\sqrt{x^2 + y^2 + z^2}.

What is a unit vector used for?
To fix a direction without any length attached.

Can a vector be negative?
Yes, but its magnitude is never negative.

What's the difference between parallel and collinear vectors?
Parallel vectors share direction but can sit on different lines; collinear vectors are parallel and lie on the same line.