Magnitude & Direction

• Pythagorean Foundation: Because the horizontal ($i$) and vertical ($j$) components of a 2D vector are perpendicular, they form a right-angled triangle where the vector itself is the hypotenuse. Therefore, the magnitude is calculated as $|\mathbf{a}| = \sqrt{x^2 + y^2}$.

• Trigonometric Orientation: The direction angle $\theta$ is derived from the ratio of the components. Using the tangent function, $\tan(\theta) = \frac{y}{x}$, which allows the angle to be found via the inverse tangent function.

• Scalar Multiples: Multiplying a vector by a positive scalar $k$ changes its magnitude by a factor of $k$ but leaves its direction unchanged. Multiplying by a negative scalar reverses the direction (180-degree rotation).

Calculating Magnitude

• Step 1: Identify the $x$ and $y$ components of the vector.
• Step 2: Square both components (note that negative signs disappear during squaring).
• Step 3: Sum the squares and take the square root of the result.

Finding the Unit Vector

• A unit vector is a vector with a magnitude of exactly 1 that points in the same direction as the original vector.
• To find it, divide the original vector by its own magnitude: $\mathbf{\hat{a}} = \frac{\mathbf{a}}{|\mathbf{a}|}$.

Resolving into Components

• If the magnitude $r$ and direction $\theta$ are known, the vector can be written in component form as:

$\mathbf{a} = (r \cos \theta)\mathbf{i} + (r \sin \theta)\mathbf{j}$

• Distance vs. Displacement: Magnitude represents the distance between two points, whereas the vector itself represents the displacement (the journey including the direction).

• Exact Values: In exams, always provide the magnitude in simplified surd form unless a decimal approximation is specifically requested. For example, use $\sqrt{50} = 5\sqrt{2}$ rather than $7.07$.

• Quadrant Awareness: When calculating the direction angle using $\tan^{-1}(\frac{y}{x})$, your calculator will only return values in a specific range. Always draw a quick sketch to determine which quadrant the vector is in and adjust the angle accordingly (e.g., add $180^{\circ}$ if the vector points into the third quadrant).

• Unit Vector Verification: After calculating a unit vector, you can verify your work by checking its magnitude; it must equal exactly 1.

• Notation Check: Ensure you use the correct notation for magnitude ($|\mathbf{a}|$) and underline your vectors in handwritten work to distinguish them from scalars.