Vectors

5 minQuiz at the end

What is a Vector?

A vector is a quantity with both magnitude (size) and direction. Unlike scalars (which only have size), vectors represent things like displacement, velocity, and force.

Notation: a, a, or column vector (x, y)

Column Vectors

A vector from A to B is written as: AB = (3, 4) means 3 right, 4 up.

Adding Vectors

Add the corresponding components:

  • (2, 3) + (-1, 5) = (1, 8)

Geometrically: place vectors head to tail.

Scalar Multiplication

Multiply each component by the scalar:

  • 3 Γ— (4, -2) = (12, -6)

A negative scalar reverses the direction.

Magnitude

The magnitude (length) of vector (x, y): |v| = √(x² + y²)

|(3, 4)| = √(9 + 16) = √25 = 5

Parallel Vectors

Two vectors are parallel if one is a scalar multiple of the other:

  • (2, 4) and (1, 2) are parallel (Γ— Β½)

Equal Vectors

Two vectors are equal if they have the same magnitude and direction (position doesn't matter).

Position Vectors

A position vector describes a point's location from the origin O:

  • OA = (3, 4) means point A is at (3, 4)