In vectors, magnitude and direction are key properties:

1. Magnitude: The length or size of a vector, denoted $|\mathbf{a}|$ . For a vector $\mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix}$in 2D, the magnitude is:

In 3D, for $\mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}$:

2. Direction: The angle the vector makes with a reference axis, often calculated using trigonometry. In 2D, if $\theta$ is the angle with the positive $x$-axis:

Magnitude tells us how long the vector is, while direction indicates where it points. Together, they fully describe the vector's properties.

📑Example 1: Find the Direction of the Vector $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$

• Diagram Representation: The vector is represented as going from the origin to the point (3, 2).

• Calculation:

📑Example 2: Find the Direction of the Vector $\begin{pmatrix} -4 \\ -2 \end{pmatrix}$

• Diagram Representation: The vector is represented as going from the origin to the point $(-4, -2).$

• Calculation:

Since the vector is in the second quadrant, we adjust the angle:

📑Example 1: Find the Magnitude of $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$

• Diagram Representation: The vector is represented as going from the origin to the point $(2, 3)$.

• Calculation:

📑Example 2: Find the Magnitude of $\mathbf{x}$ when $\mathbf{x} = \begin{pmatrix} -2 \\ -4 \end{pmatrix}$

• Note: This could be worded as "find $|\mathbf{x}|$": The bars mean "magnitude".
• Calculation:

📑Example: Write in the form $xi + yj$ the vector with direction 120° and magnitude $3$.

Diagram Representation:

• A right triangle is drawn with the angle $\alpha$ and sides labelled as opposite (OPP), adjacent (ADJ), and hypotenuse (HYP). Notes:

1. Trigonometric Relations:

• $\sin \theta = \frac{\text{OPP}}{\text{HYP}}$
• Therefore, $\sin \theta = \frac{\text{OPP}}{l}$ which implies $\text{OPP} = l \sin \theta$.
• $\cos \theta = \frac{\text{ADJ}}{\text{HYP}}$
• Therefore, $\cos \theta = \frac{\text{ADJ}}{l}$ which implies $\text{ADJ} = l \cos \theta$.

3. Horizontal and Vertical Components:

• This approach extends to any horizontal and vertical components. Calculations:

• For the $x$-component:

• For the $y$-component:

• Vector Representation: