Vectors

Study Notes

Vectors are mathematical objects that have both magnitude and direction. They can be added, subtracted, and multiplied by scalars, but not divided.

• Vector — a quantity with both magnitude and direction. Example: The vector $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ represents a movement 3 units right and 4 units up.
• Scalar — a real number that can multiply a vector. Example: Multiplying the vector $\begin{pmatrix} 2 \\ 5 \end{pmatrix}$ by 3 gives $\begin{pmatrix} 6 \\ 15 \end{pmatrix}$.
• Column Vector — a way to represent vectors using vertical arrays. Example: $\begin{pmatrix} x \\ y \end{pmatrix}$ represents a vector in 2D space.
• Position Vector — a vector that starts at the origin. Example: The position vector $\begin{pmatrix} 4 \\ -2 \end{pmatrix}$ points from the origin to the point (4, -2).
• Magnitude — the length of a vector. Example: The magnitude of $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ is 5.

Exam Tips

Key Definitions to Remember

• A vector has both magnitude and direction.
• A scalar is a real number used to multiply a vector.
• A position vector starts at the origin.

Common Confusions

• Confusing vectors with scalars, which only have magnitude.
• Forgetting that vectors cannot be divided.

Typical Exam Questions

• What is the magnitude of $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$? The magnitude is 5.
• How do you add vectors $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 4 \end{pmatrix}$? The result is $\begin{pmatrix} 3 \\ 7 \end{pmatrix}$.
• What is the result of multiplying $\begin{pmatrix} 2 \\ 5 \end{pmatrix}$ by 3? The result is $\begin{pmatrix} 6 \\ 15 \end{pmatrix}$.

What Examiners Usually Test

• Ability to perform vector addition and subtraction.
• Calculating the magnitude of a vector.
• Understanding and using position vectors.