Equality of Vectors
What is a Vector?
Before we discuss equality, let's recall what a vector is:
A vector is a mathematical object that has both magnitude (length) and direction.
We denote vectors as $\vec{AB}$, $\vec{a}$, or in bold as a. A vector $\vec{AB}$ starts at point $A$ (the tail) and ends at point $B$ (the head).
📐 Definition: Equality of Vectors
Two vectors are equal if and only if they have the same magnitude AND the same direction.
Mathematically, $\vec{a} = \vec{b}$ if and only if:
- Same Magnitude: $|\vec{a}| = |\vec{b}|$ (their lengths are equal)
- Same Direction: Both vectors point in exactly the same direction (they are parallel and have the same sense)
Understanding with Coordinates
If vector $\vec{a}$ has components $(a_1, a_2)$ and vector $\vec{b}$ has components $(b_1, b_2)$, then:
$$\vec{a} = \vec{b} \iff a_1 = b_1 \text{ and } a_2 = b_2$$
For a vector from point $A(x_1, y_1)$ to point $B(x_2, y_2)$:
$$\vec{AB} = (x_2 - x_1,\; y_2 - y_1)$$
The magnitude is: $|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
⚡ Common Misconceptions
| ❌ Wrong Thinking | ✅ Correct Understanding |
|---|---|
| Vectors must start at the same point to be equal | Position doesn't matter — only magnitude and direction |
| Vectors pointing in opposite directions with same length are equal | They must have the same direction. Opposite direction means $\vec{a} = -\vec{b}$ |
| $\vec{AB} = \vec{BA}$ | $\vec{AB} = -\vec{BA}$ (they have opposite directions!) |