Vectors

Study Notes

Vectors are mathematical objects that have both magnitude and direction and can be used to represent quantities like force and velocity. They can be added, subtracted, and multiplied by scalars to solve geometric problems.

• Vector — a quantity with both magnitude and direction. Example: Displacement is a vector.
• Scalar — a quantity with magnitude but no direction. Example: Temperature is a scalar.
• Column Vector — a way to represent vectors using vertical components. Example: $\begin{pmatrix} 5 \\ 3 \end{pmatrix}$ represents a vector with horizontal component 5 and vertical component 3.
• Magnitude of a Vector — the length of the vector. Example: Calculated using Pythagoras’ Theorem for a right-angled triangle.
• Position Vector — a vector that starts at the origin. Example: $\begin{pmatrix} x \\ y \end{pmatrix}$ represents a position vector from the origin to point (x, y).

Exam Tips

Key Definitions to Remember

• A vector has both magnitude and direction.
• A scalar has only magnitude.
• Column vectors represent vectors using vertical components.

Common Confusions

• Confusing vectors with scalars.
• Forgetting that vectors can be added or subtracted but not divided.

Typical Exam Questions

• How do you add two vectors?
  Use the nose-to-tail method or add corresponding components.
• What is the magnitude of vector $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$?
  5, using Pythagoras’ Theorem.
• How do you represent a vector in column form?
  Use the horizontal and vertical components.

What Examiners Usually Test

• Ability to add and subtract vectors.
• Calculating the magnitude of vectors.
• Understanding and using position vectors.