Scalars and Vectors

🧭 Plot Summary

Every quantity you'll use this year is either a scalar — just a number — or a vector — a number and a direction. Distance and speed are scalars; position, displacement, velocity, and acceleration are vectors, and position even comes with its own named vector, r, whose direction is captured by the unit vector r̂. You can represent any vector as an arrow — its length shows magnitude, its arrowhead shows direction — or write it in unit vector notation, breaking it into independent x-, y-, and z-components using î, ĵ, and k̂. Add two or more vectors by adding their components separately, and the sum is the resultant. In one dimension, you don't even need an arrow — a plus or minus sign carries the whole direction.

What you'll do in this lesson

• Distinguish scalars (magnitude only) from vectors (magnitude and direction).
• Represent vectors as arrows — length shows magnitude, arrowhead shows direction.
• Classify distance, speed, position, displacement, velocity, and acceleration correctly.
• Write vectors in unit vector notation using î, ĵ, and k̂ for the x-, y-, and z-directions.
• Convert between component form and magnitude-and-direction form.
• Find a resultant vector by summing the components of the addend vectors.
• Describe a position vector r and the unit vector r̂ that points in its direction.
• Use positive and negative signs to encode direction in one-dimensional problems.

Why it matters

Unit vector notation is the mathematical backbone of this entire course. Once a vector is broken into components, each direction becomes its own independent one-dimensional problem — exactly how you'll solve projectile motion in Topic 1.5, forces in Unit 2, and nearly everything after. Get comfortable with î, ĵ, and k̂ now, and the rest of the year gets easier.

✅ Self-Check Before You Roll On

Check off each item as you get there. These aren't grades — they're your own signal.