Working with Vectors

A vector is a quantity that possesses both a numerical size (magnitude) and a specific direction in space. In mechanics, this distinguishes vectors from scalars, such as mass or time, which only have magnitude.

Vectors are commonly represented using two notations: column vectors $\begin{pmatrix} x \\ y \end{pmatrix}$ and unit vector notation $x\mathbf{i} + y\mathbf{j}$.

The unit vectors $\mathbf{i}$ and $\mathbf{j}$ represent a single unit of displacement in the positive horizontal ($x$) and positive vertical ($y$) directions, respectively. In written work, these are typically underlined ($\underline{i}, \underline{j}$) to denote their vector status.

Calculating Magnitude: The magnitude (length) of a vector $\mathbf{v} = x\mathbf{i} + y\mathbf{j}$ is found using the Pythagorean theorem: $|\mathbf{v}| = \sqrt{x^2 + y^2}$. This value is always a non-negative scalar.

Determining Direction: The angle $\theta$ a vector makes with the horizontal can be found using trigonometry: $\tan(\theta) = \frac{|y|}{|x|}$. It is vital to sketch the vector first to identify which quadrant it occupies.

Bearings: In navigation and mechanics, directions are often given as bearings—angles measured clockwise from North ($000^{\circ}$). The unit vector $\mathbf{j}$ corresponds to North ($000^{\circ}$) and $\mathbf{i}$ corresponds to East ($090^{\circ}$).

Resultant vs. Equilibrium: A resultant is the 'net' effect of multiple vectors, whereas equilibrium is the specific state where that net effect is zero.

Distance vs. Displacement: Distance is a scalar representing total path length; displacement is a vector representing the straight-line change in position from start to end.

Notation Accuracy: Always underline $\mathbf{i}$ and $\mathbf{j}$ in handwritten exams. Failing to distinguish vectors from scalars can lead to conceptual errors and lost marks.

Exact Values: If a question asks for an 'exact value' for magnitude, leave your answer in surd form (e.g., $\sqrt{13}$) rather than a decimal approximation.

The Sketch Rule: Always draw a quick right-angled triangle when calculating bearings or resolving components. This prevents 'calculator errors' where the inverse tangent function returns an angle in the wrong quadrant.

Sanity Check: Ensure the magnitude is always greater than or equal to the individual $x$ or $y$ components. If your calculated magnitude is smaller than a component, you have likely made an arithmetic error.

Negative Components: When squaring a negative component to find magnitude, remember that $(-3)^2 = 9$. A common mistake is entering $-3^2$ into a calculator, which results in $-9$ and an incorrect magnitude.

Bearing Confusion: Students often calculate the angle from the $x$-axis and forget to convert it to a bearing from North. Always remember: Bearing = $90^{\circ} - \text{angle from x-axis}$ (if in the first quadrant).

Mixing Units: Ensure all vectors in a sum use the same units (e.g., all Newtons for force) before calculating the resultant.