Vector Basics: quick revision, 14 bites

What is a Vector?

Magnitude and direction together.

A **vector** is a mathematical quantity that has both **magnitude** (size) and **direction**. This distinguishes it from a **scalar**, which has only magnitude. For example, velocity is a vector (e.g., 60 km/h north), while speed is a scalar (e.g., 60 km/h).

Takeaway. Vectors always include both size and direction; scalars have only size.

Vector Notation

How we write vectors.

Vectors are often written in **bold** (e.g., **v**) or with an arrow above the letter (e.g., $\vec{v}$). In print, bold is common; in handwriting, the arrow is used. The magnitude of vector **v** is written as $|\vec{v}|$ or simply $v$.

Takeaway. Vectors are denoted by bold or arrow notation; magnitude uses absolute value bars.

Common Vector Quantities

Vectors are everywhere in physics.

Takeaway. Displacement, velocity, acceleration, and force are all vector quantities.

Vector vs. Scalar Example

Walking 3 km east vs. 3 km.

If you walk 3 km east, your **displacement** is 3 km east (a vector). If you then walk 4 km north, your total displacement is not 7 km; it's the straight-line distance from start to finish. The total **distance** traveled (scalar) is 7 km.

Takeaway. Distance is scalar; displacement is vector and may be less than the path length.

Adding Vectors Graphically

Tip-to-tail method.

Takeaway. Graphical addition uses tip-to-tail placement; the resultant closes the polygon.

Component Form of a Vector

Breaking vectors into parts.

A vector $\vec{v}$ in 2D can be expressed as $\vec{v} = (v_x, v_y)$, where $v_x$ and $v_y$ are the **components** along the x- and y-axes. The magnitude is found using the Pythagorean theorem.

Takeaway. Components allow algebraic manipulation of vectors.

Adding Vectors by Components

Algebraic addition is precise.

Takeaway. Component addition gives exact resultant magnitude and direction.

Forgetting Direction in Addition

Direction matters in sums.

Takeaway. Vector addition must account for direction, not just magnitude.

Navigation and GPS

Vectors guide your route.

GPS devices use vector addition to compute the shortest path (displacement) between two points. They account for direction changes and calculate the resultant vector to guide you. Without vectors, navigation would be impossible.

Takeaway. GPS relies on vector mathematics to find efficient routes.

Walking on a Grid

Like moving on a chessboard.

Imagine a chessboard: moving 3 squares right and 4 squares up is like a vector (3,4). The straight-line distance (magnitude) is 5 squares. The path you actually walk (3+4=7) is the scalar distance. Vectors capture the net effect.

Takeaway. Vectors give the net change, not the path length.

Check Your Quadrant

Angle sign matters.

Takeaway. Always verify the quadrant when finding direction from components.

SOH CAH TOA for Components

Trig for vector parts.

To remember how to find components: **SOH** CAH TOA. For a vector at angle $\theta$ from the x-axis: $v_x = |\vec{v}| \cos \theta$ (adjacent), $v_y = |\vec{v}| \sin \theta$ (opposite). The hypotenuse is the magnitude.

Takeaway. Use cosine for x-component, sine for y-component.

Vectors in 3D

Vectors aren't just 2D.

Vectors can exist in three dimensions, written as $\vec{v} = (v_x, v_y, v_z)$. The magnitude is $|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$. 3D vectors are essential in physics for describing forces, fields, and motion in space.

Takeaway. Vectors extend naturally to 3D with three components.

Always Sketch First

Draw before you calculate.

Before doing any vector calculation, sketch the vectors roughly to scale. This gives you a visual check for the resultant direction and helps avoid sign errors. A quick drawing can confirm whether your answer makes sense.

Takeaway. A rough sketch prevents many common vector mistakes.