Physics Vectors and Graphs Study Guide

What Vectors Actually Are (And Why Direction Matters)

Scalar quantities have magnitude only. Temperature, mass, time — these are scalars. You state a number and you're done.

Vectors have magnitude and direction. Velocity, force, displacement — these need both a number and a direction to mean anything. Saying "I'm walking at 5 m/s" is incomplete. Saying "I'm walking 5 m/s north" tells you something useful.

This distinction matters more than most students realize. Physics problems fail here first.

Vector Representation

You'll see vectors written a few ways:

• Arrow notation: →v means "the vector v"
• Component form: (3, 4) means 3 units in x-direction, 4 units in y-direction
• Magnitude-direction form: |v| = 10 at 45° means the vector is 10 units long at a 45-degree angle

Get comfortable switching between these. Exams will present information in one form and expect you to convert it.

Vector Operations You Need to Know

Addition and Subtraction

Add vectors by placing them head-to-tail. The resultant vector goes from the start of the first to the end of the last.

For subtraction, reverse the direction of the vector you're subtracting, then add.

Graphically, this means drawing parallelograms or tip-to-tail constructions. Numerically, add the x-components together and the y-components together.

Scalar Multiplication

Multiplying a vector by a scalar changes its magnitude but keeps its direction (unless the scalar is negative, which flips the direction).

2 × (3, 4) = (6, 8)

The vector gets longer. Negative 2 × (3, 4) = (-6, -8). The vector flips.

Dot Product vs. Cross Product

Students confuse these constantly.

Dot product gives a scalar. A · B = |A||B|cos(θ). Used for work, finding angles between vectors.

Cross product gives a vector. A × B = |A||B|sin(θ) in a direction perpendicular to both. Used for torque, magnetic forces.

Know which one your problem requires. The wording usually makes it clear — "find the work done" means dot product, "find the torque vector" means cross product.

Breaking Vectors Into Components

This is where most vector problems get solved. You take a vector at an angle and split it into x and y parts.

If you have a vector with magnitude v at angle θ from the horizontal:

• v_x = v cos(θ)
• v_y = v sin(θ)

Example: A 10 N force pulls at 30° above the horizontal.

• F_x = 10 × cos(30°) = 10 × 0.866 = 8.66 N
• F_y = 10 × sin(30°) = 10 × 0.5 = 5 N

Why does this matter? Because forces in perpendicular directions don't affect each other. You can solve x and y components independently, then combine results.

Graphs in Physics — The Basics

Physics graphs aren't just pictures. They're mathematical relationships you can extract data from.

The key rule: the slope of a graph equals the rate of change. The area under a graph has its own meaning too.

Position vs. Time Graphs

Slope = velocity

A straight line means constant velocity. A curved line means changing velocity — the slope at any point gives the instantaneous velocity there.

Horizontal line means zero velocity. You're not moving.

Velocity vs. Time Graphs

Slope = acceleration

Area under the graph = displacement

This one catches students. The area between the velocity curve and the x-axis gives you the distance traveled. If the graph dips below zero, that section subtracts from your total displacement.

Acceleration vs. Time Graphs

Slope = jerk (rate of change of acceleration)

Area under the graph = change in velocity

These come up less often but show up in higher-level motion problems.

Reading Motion Graphs — Common Patterns

How To: Solve Vector Problems Step by Step

Most vector problems follow the same pattern. Here's how to work through them:

1. Draw a diagram. Always. Even if it's rough. A good diagram prevents half your mistakes.
2. Identify knowns and unknowns. Write down what you know about magnitudes and directions.
3. Choose a coordinate system. Usually x = horizontal, y = vertical. Align axes with your problem's geometry.
4. Break diagonal vectors into components. Use sin and cos based on the angle given.
5. Set up equations for each direction. Apply Newton's laws or kinematic equations in x and y separately.
6. Solve algebraically. Keep symbols as long as possible. Plug numbers in at the end.
7. Check your work. Does the magnitude make sense? Is the direction reasonable?

Common Mistakes That Cost Points

• Confusing position with displacement, or velocity with speed
• Forgetting that negative velocity means direction, not "backwards speed"
• Measuring angles from the wrong axis
• Adding vectors that aren't tip-to-tail
• Ignoring significant figures
• Using degrees instead of radians in calculator (or vice versa)
• Forgetting units on final answers

The angle mistake is the most common. If a problem says "30° above the horizontal," measure from horizontal. If it says "30° from the wall," measure from the wall. Read carefully.

Quick Reference: Key Formulas

Getting Started: What to Practice Tonight

If you're cramming for an exam tomorrow:

• Practice resolving vectors into components — 10 problems minimum
• Draw position-time and velocity-time graphs from motion descriptions
• Extract velocity from position-time graphs by drawing tangent lines and calculating slopes
• Calculate displacement from velocity-time graphs by counting grid squares under the curve

Don't try to memorize everything. Understand the relationships. If you know slope equals rate of change and area equals accumulated quantity, you can reconstruct most of physics from those two ideas.