Coordinates of a Vector
What is a Vector?
A vector is a mathematical object that has both magnitude (length) and direction. Unlike a simple number (scalar), a vector tells us how far and which way.
We represent vectors visually as arrows — the arrow's length shows the magnitude, and the arrow's direction shows where it points.
Formal Definition: Coordinates of a Vector
If we write $\vec{a} = (a_x, a_y)$, then:
- $a_x$ is the x-component (horizontal displacement)
- $a_y$ is the y-component (vertical displacement)
The coordinates tell us: move $a_x$ units horizontally and $a_y$ units vertically to get from the start to the end of the vector.
Key Properties
Magnitude (Length)
The magnitude of $\vec{a} = (a_x, a_y)$ is:
$|\vec{a}| = \sqrt{a_x^2 + a_y^2}$
Equal Vectors
Two vectors are equal if and only if their corresponding coordinates are equal:
$\vec{a} = \vec{b} \iff a_x = b_x$ and $a_y = b_y$
Zero Vector
$\vec{0} = (0, 0)$ — has no direction and zero magnitude.
Opposite Vector
If $\vec{a} = (a_x, a_y)$, then $-\vec{a} = (-a_x, -a_y)$
Operations with Vector Coordinates
| Operation | Formula | Example |
|---|---|---|
| Addition | $\vec{a} + \vec{b} = (a_x + b_x,\; a_y + b_y)$ | $(3, 2) + (1, 4) = (4, 6)$ |
| Subtraction | $\vec{a} - \vec{b} = (a_x - b_x,\; a_y - b_y)$ | $(3, 2) - (1, 4) = (2, -2)$ |
| Scalar Multiplication | $k\vec{a} = (k \cdot a_x,\; k \cdot a_y)$ | $3 \cdot (2, -1) = (6, -3)$ |
Notation Guide
- $\vec{a}$ — vector a
- $\vec{AB}$ — vector from A to B
- $(a_x, a_y)$ — coordinates
- $|\vec{a}|$ — magnitude