Introduction

Vectors are one of the first truly “new” mathematical objects we meet after basic algebra. They let us describe movement, direction, and quantities with more than one component. This article introduces vectors in two friendly ways:

• as arrows you can draw;
• as lists of numbers you can calculate with.

What Is a Vector?

A vector is a quantity that has both:

• a direction (which way it points), and
• a magnitude (how long it is).

Examples of things that behave like vectors:

• a push or pull (a force),
• a movement from one place to another,
• a wind blowing in a certain direction.

A vector is not just a number like $3$ or $-7$. Those are scalars—they have size but no direction.

Vectors as Arrows in Space

A simple way to picture a vector is as an arrow.

• The tail of the arrow is where the vector starts.
• The head of the arrow is where it points.
• The length of the arrow shows the magnitude.
• The orientation shows the direction.

Key ideas:

• You can draw the same vector starting anywhere. Only the difference between start and end matters.
• Two arrows with the same length and direction represent the same vector, even if they are drawn in different places.

Example:
An arrow going 3 units to the right and 2 units up represents the vector “right 3, up 2”.

Vectors as Lists of Numbers

Instead of drawing arrows, we can describe vectors using coordinates.

• In 2D, a vector is written as $$(a, b)$$ meaning “$a$ units in the $x$-direction, $b$ units in the $y$-direction.”
• In 3D, a vector is $$(a, b, c)$$ meaning “$a$ units in $x$, $b$ in $y$, $c$ in $z$.”

Examples:

• $(3, 2)$ means “right 3, up 2.”
• $(-1, 4, 0)$ means:
  • left 1
  • up 4
  • no movement in the $z$-direction.

These lists of numbers match the arrows:
Start at the origin $(0,0)$, move according to the numbers, and you reach the arrow’s head.

Length and Direction

The length (or magnitude) of a vector $(a, b)$ in 2D comes from the Pythagorean theorem: $$\| (a, b) \| = \sqrt{a^2 + b^2}.$$ Examples:

• $\|(3,4)\| = \sqrt{3^2 + 4^2} = 5$.
• $\|(1,1)\| = \sqrt{2}$.

The direction of a vector can be thought of as the angle it makes with the horizontal axis. You don’t need trigonometry yet—just the idea that:

• longer vectors point the same way as shorter ones if their components are proportional;
• e.g., $(2,4)$ and $(1,2)$ point in the same direction.

Visualising Vectors in Two Dimensions

In 2D, vectors live on a flat plane.

Ways to picture them:

• Draw a grid (like graph paper).
• Start at the origin $(0,0)$.
• Move right/left according to the first number.
• Move up/down according to the second number.
• Mark the endpoint and draw an arrow.

Helpful observations:

• Vectors with the same direction lie on the same line through the origin.
• Vectors with the same length lie on a circle around the origin.

Visualising Vectors in Three Dimensions

3D vectors add a third component, which is harder to draw but follows the same idea.

To visualise:

• Imagine the usual 2D grid as the floor.
• The third component tells you how far “up” or “down” to go.
• Draw an arrow from the origin to the point $(a,b,c)$.

Key points:

• 3D vectors describe movement in space—useful for physics, engineering, and computer graphics.
• Even if drawing is tricky, the list-of-numbers view works perfectly.

Vector Notation

Vector notation gives us compact, readable ways to describe a vector’s direction and length. Three of the most common forms are:

• Arrow notation (the “arrow over a letter”)
• Component notation (using coordinates like $(x, y)$)
• Angle + length notation (sometimes called polar form)

Each form highlights something different, and being able to convert between them is a core skill.

Arrow Notation

Arrow notation uses a symbol like $\vec{v}$ or $\overrightarrow{AB}$.

• $\vec{v}$ means “vector $v$” without specifying where it starts.
• $\overrightarrow{AB}$ means “the vector from point $A$ to point $B$.”

Key ideas:

• The arrow tells you it’s a vector, not a number.
• Length and direction are built into the meaning, even if not written explicitly.
• This notation is great for diagrams and geometric reasoning.

Example:

• $\overrightarrow{AB}$ is the arrow from $A(1,2)$ to $B(4,6)$.

Component Notation

Component notation writes a vector as: $$(x, y)$$ This means:

• Move $x$ units horizontally (positive = right, negative = left)
• Move $y$ units vertically (positive = up, negative = down)

Advantages:

• Very easy to add and subtract vectors.
• Works well for algebraic calculations.

Example:

• $\vec{v} = (3, -2)$ means “3 units right, 2 units down.”

Angle + Length Notation (Polar Form)

A vector can also be described by:

• Its length (also called magnitude), written $|\vec{v}|$
• Its direction angle $\theta$, measured from the positive $x$-axis

Written as: $$(|\vec{v}|,\ \theta)$$ Example:

• A vector of length $5$ at a $30^\circ$ angle can be written as $(5,\ 30^\circ)$.

This form is especially useful when direction is the main focus.

Converting Between Component Form and Angle+Length Form

From Components $(x, y)$ to Length + Angle

1. Length: $$|\vec{v}| = \sqrt{x^2 + y^2}$$
2. Angle: $$\theta = \tan^{-1}\!\left(\frac{y}{x}\right)$$ (Adjust the angle depending on the quadrant.)

Example: $$\vec{v} = (3, 4)$$

• Length: $\sqrt{3^2 + 4^2} = 5$
• Angle: $\tan^{-1}(4/3) \approx 53.1^\circ$

So the vector is $(5,\ 53.1^\circ)$.

From Length + Angle to Components

If a vector has length $L$ and angle $\theta$: $$x = L\cos\theta$$ $$y = L\sin\theta$$ Example:
Vector of length $10$ at $30^\circ$:

• $x = 10\cos 30^\circ = 5\sqrt{3}$
• $y = 10\sin 30^\circ = 5$

So the vector is $(5\sqrt{3},\ 5)$.

Converting Between Arrow Notation and Components

Arrow notation $\overrightarrow{AB}$ can be turned into components by subtracting coordinates:

If $A = (x_1, y_1)$ and $B = (x_2, y_2)$: $$\overrightarrow{AB} = (x_2 - x_1,\ y_2 - y_1)$$ Example: $$A(1,2), (4,6)$$ $$\overrightarrow{AB} = (4-1,\ 6-2) = (3,4)$$ Once you have components, you can convert to angle+length form as above.

Common Misconceptions and How to Avoid Them

• “A vector is just a point.”
  A point is a location; a vector is a movement. They look similar on a graph but mean different things.
• “The starting point matters.”
  It doesn’t. Only the difference between start and end matters.
• “Vectors must be in 2D or 3D.”
  Not true—vectors can have any number of components, like $(a,b,c,d)$. We just can’t draw them easily.
• “Vectors are always physical.”
  Many vectors represent abstract things, like changes in data or directions in high‑dimensional spaces.

Summary

• A vector has magnitude and direction.
• You can picture vectors as arrows or as lists of numbers.
• The length of $(a,b)$ is $\sqrt{a^2 + b^2}$.
• Vectors describe movement, forces, and many real‑world quantities.
• The starting point of an arrow does not matter—only its direction and length.

Calculator

Exercises

1. Draw the vector $(4,1)$ on graph paper. What direction does it point?

   Solution

   Vector $(4,1)$ direction

   • From the origin, move 4 units right and 1 unit up.
   • It points mostly to the right, slightly upward.
2. Which of the following vectors point in the same direction as $(2,3)$?
   • $(4,6)$, $(1,1)$, $(-2,-3)$.

   Solution

   Same direction as $(2,3)$

   • Vectors with the same direction are scalar multiples of each other.
   • $(4,6) = 2(2,3)$ → same direction.
   • $(1,1)$ is not a multiple of $(2,3)$ → different direction.
   • $(-2,-3) = -1(2,3)$ → opposite direction (same line, but reversed).
3. Find the length of each vector:
   • $(3,4)$, $(5,12)$, $(1,2)$.

   Solution

   Lengths

   • $\|(3,4)\| = \sqrt{3^2 + 4^2} = \sqrt{9+16} = \sqrt{25} = 5$.
   • $\|(5,12)\| = \sqrt{5^2 + 12^2} = \sqrt{25+144} = \sqrt{169} = 13$.
   • $\|(1,2)\| = \sqrt{1^2 + 2^2} = \sqrt{1+4} = \sqrt{5}$.
4. Describe in words what the vector $(-3,2)$ means.

   Solution

   Meaning of $(-3,2)$

   • Move 3 units left (negative $x$) and 2 units up (positive $y$).
   • So it represents a movement “left 3, up 2”.
5. True or false: “The vectors $(1,2)$ and $(2,1)$ have the same direction.”

   Solution

   Same direction? $(1,2)$ and $(2,1)$

   • Check if one is a scalar multiple of the other.
   • There is no number $k$ such that $k(1,2) = (2,1)$.
   • So the statement is false.
6. Give one real‑world example of a vector and explain why it is a vector.

   Solution

   Real‑world vector example

   • Example: wind velocity “10 km/h to the north”.
   • It has size (10 km/h) and direction (north), so it is a vector.
7. Which of these are scalars and which are vectors?
   • Temperature, velocity, mass, force.

   Solution

   Scalars vs vectors

   • Temperature → scalar (size only).
   • Velocity → vector (speed and direction).
   • Mass → scalar.
   • Force → vector (magnitude and direction).
8. A vector has length $10$ and points straight upward. Write a possible coordinate form for it.

   Solution

   Vector of length 10 pointing straight up

   • In 2D, “straight up” means no $x$-movement, only $y$-movement.
   • A suitable vector is $$(0,10).$$
9. Explain why the point $(3,4)$ and the vector $(3,4)$ are not the same thing.

   Solution

   Why $(3,4)$ as a point vs as a vector differ

   • As a point, $(3,4)$ means “the location 3 units right and 4 units up from the origin.”
   • As a vector, $(3,4)$ means “a movement of 3 right and 4 up,” which can start anywhere.
   • Same coordinates, but different meaning: location vs displacement.