Vectors in 2D

Summary

Vectors in 2D involve understanding both the direction and magnitude of vectors, as well as operations like addition, subtraction, and scalar multiplication.

• Vector Notation — Vectors are expressed using starting and ending points with an arrow or as column vectors. Example: Vector AB can be written as $\begin{pmatrix} 6 \\ 4 \end{pmatrix}$.
• Magnitude — The length of a vector, calculated using its components. Example: For vector $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$, the magnitude is 5.
• Directed Line Segment — A vector with a specific initial and terminal point. Example: A vector from point A to B.
• Translation by Vector — Moving a point by a vector's components. Example: Moving a point by (-3, 1) means 3 units left and 1 unit up.
• Sum of Vectors — Adding vectors to find a resultant vector. Example: Adding $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ gives $\begin{pmatrix} 4 \\ 6 \end{pmatrix}$.
• Scalar Multiplication — Multiplying a vector by a scalar changes its magnitude. Example: Multiplying $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$ by 2 gives $\begin{pmatrix} 4 \\ 6 \end{pmatrix}$.

Exam Tips

Key Definitions to Remember

• Vector: A quantity with both magnitude and direction.
• Magnitude: The length of a vector.
• Scalar Multiplication: Multiplying a vector by a number.

Common Confusions

• Confusing direction with magnitude.
• Forgetting to change direction when multiplying by a negative scalar.

Typical Exam Questions

• What is the magnitude of vector $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$? Answer: 5
• How do you express vector AB in column form if it moves 6 units right and 4 units up? Answer: $\begin{pmatrix} 6 \\ 4 \end{pmatrix}$
• What happens to a vector when multiplied by -2? Answer: Its direction reverses and magnitude doubles.

What Examiners Usually Test

• Understanding of vector notation and representation.
• Ability to calculate magnitude and perform vector addition.
• Application of scalar multiplication and its effects on vectors.