Vector representation

Drawing vectors

A vector is drawn as an arrow: a directed line segment.

• The tail is the starting point.
• The head (arrowhead) is the end.
• The length shows the magnitude (to scale).
• The arrow shows the direction.

Naming vectors

AB⃗ and BA⃗ have the same length but opposite directions: BA⃗ = −AB⃗.

Scale diagrams

Choose a scale and draw carefully.

Example: Draw 40 N at 30° above the horizontal with a scale of 1 cm = 10 N.

• Draw a 4 cm arrow at 30° from a horizontal reference line.

Component form

A vector can be written by how far it moves horizontally and vertically: component form.

v = [3, 4] (or ⟨3, 4⟩ or (3, 4))

means 3 units right and 4 units up.

From two points: A(1, 2) to B(6, −1)

AB⃗ = [6 − 1, −1 − 2] = [5, −3]

(Subtract tail from head.)

Position doesn’t matter

Vectors with the same magnitude and direction are equal, wherever they’re drawn. [5, −3] starting at (0, 0) is the same vector as [5, −3] starting at (10, 7).

In three dimensions

v = [x, y, z], e.g., [2, −1, 4]: 2 along x, −1 along y, 4 along z.

Unit vectors

A unit vector has magnitude 1. Special ones:

So [3, 4] = 3i + 4j, and [2, −1, 4] = 2i − j + 4k.

Converting between forms

Worked examples

Example 1: Vector from P(−2, 3) to Q(4, 7)? [6, 4].

Example 2: Write [−5, 2] using i and j. −5i + 2j.

Example 3: Is [2, 3] from (0, 0) equal to [2, 3] from (5, 5)? Yes.

Common mistakes

• Subtracting head from tail (wrong sign).
• Forgetting that position doesn’t change a vector.
• Mixing up point coordinates and vector components.

Try it yourself

1. Vector from A(3, −1) to B(−2, 4)?
2. Write 4i − 7j in component form.
3. Vector from C(1, 0, 2) to D(4, 5, −1)?
4. What is BA⃗ if AB⃗ = [2, −6]?

Answers: 1) [−5, 5] 2) [4, −7] 3) [3, 5, −3] 4) [−2, 6]

Quick recap

• A vector is a directed line segment from tail to head.
• Component form [x, y] = head − tail.
• Unit vectors i, j, k build any vector.