Working with Vectors

Vector Addition: To add two vectors, you combine their corresponding components. In column notation, this involves adding the top numbers together and the bottom numbers together; in $\mathbf{i}, \mathbf{j}$ notation, you sum the coefficients of the like terms.

The Triangle Law: Geometrically, adding vectors follows the 'nose-to-tail' rule. If you place the start (tail) of the second vector at the end (head) of the first, the resultant vector is the direct path from the start of the first to the end of the second.

Scalar Multiplication: Multiplying a vector by a scalar $k$ changes its magnitude by a factor of $|k|$. If $k$ is negative, the vector's direction is reversed, though it remains parallel to the original orientation.

Calculating Magnitude: The magnitude (or modulus) 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 represents the straight-line distance from the vector's start point to its end point.

Determining Direction: The direction is typically expressed as an angle $\theta$ relative to a horizontal or vertical axis, calculated using trigonometry: $\tan(\theta) = \frac{y}{x}$. It is essential to sketch the vector to identify which quadrant the angle falls in.

Bearings: In many contexts, direction is given as a bearing, which is a three-figure angle measured clockwise from North ($000^{\circ}$). In this system, the unit vector $\mathbf{j}$ points North and $\mathbf{i}$ points East.

Component Form: Resolving a vector means breaking it down into its horizontal ($x$) and vertical ($y$) components. If a vector has magnitude $R$ and makes an angle $\theta$ with the positive x-axis, its components are $x = R \cos(\theta)$ and $y = R \sin(\theta)$.

Applications in Mechanics: This process is vital for analyzing forces acting at angles. By resolving all forces into $\mathbf{i}$ and $\mathbf{j}$ components, you can sum them independently to find the total effect on an object.

Equilibrium: A system is in equilibrium when the resultant of all acting vectors is the zero vector $\binom{0}{0}$. This implies that the sum of all horizontal components and the sum of all vertical components must both equal zero.

Distance vs. Displacement: Distance is a scalar representing the total path length, while displacement is a vector representing the change in position from start to finish.

Speed vs. Velocity: Speed is the magnitude of the velocity vector. Two objects can have the same speed but different velocities if they are moving in different directions.

Notation Accuracy: In handwritten work, always underline vector symbols (e.g., $\underline{u}$, $\underline{i}$, $\underline{j}$) to distinguish them from scalar variables, as you cannot write in bold.

Exact Values: If a question asks for an exact magnitude, leave your answer in simplified surd form (e.g., $3\sqrt{10}$) rather than a decimal approximation.

Sketching for Bearings: Always draw a quick coordinate sketch when calculating bearings. This prevents 'quadrant errors' where you might calculate the internal angle of a triangle but fail to convert it to the clockwise-from-north standard.

Sanity Check: Ensure your resultant vector makes sense. If you add two vectors pointing generally 'up and right', your resultant should not have negative components.