Vectors

Summary

Vectors are mathematical objects that have both a magnitude and a direction, used to describe translations and construct geometric arguments. Scalars, on the other hand, have only magnitude and no direction.

• Vector — a quantity with both magnitude and direction. Example: Force, velocity, displacement.
• Scalar — a quantity with only magnitude and no direction. Example: Mass, volume, temperature.
• Column Vector — a vector represented with a top number for the horizontal component and a bottom number for the vertical component. Example: $\begin{pmatrix} 5 \\ 3 \end{pmatrix}$.
• Addition of Vectors — combining vectors by adding corresponding components or using the nose-to-tail method. Example: $a + b$.
• Multiplication of Vectors — multiplying a vector by a scalar affects its magnitude and possibly its direction. Example: $2x$ or $-3x$.
• Parallel Vectors — vectors that have the same direction and proportional components. Example: $\begin{pmatrix} 2 \\ 4 \end{pmatrix}$ and $\begin{pmatrix} 4 \\ 8 \end{pmatrix}$.
• Magnitude of a Vector — calculated using Pythagoras’ Theorem. Example: $|a|$.

Exam Tips

Key Definitions to Remember

• Vector: A quantity with both magnitude and direction.
• Scalar: A quantity with only magnitude and no direction.
• Column Vector: A vector with horizontal and vertical components.

Common Confusions

• Mixing up vectors and scalars.
• Forgetting to change direction when multiplying by a negative scalar.

Typical Exam Questions

• What is a vector? A quantity with both magnitude and direction.
• How do you add vectors? By adding corresponding components or using the nose-to-tail method.
• How do you find the magnitude of a vector? Use Pythagoras’ Theorem.

What Examiners Usually Test

• Understanding of vector addition and subtraction.
• Ability to multiply vectors by scalars.
• Calculation of vector magnitude.