Geometry · Quick Reference

Vectors (Basic) Cheat Sheet

Geometry · Lesson 16/16

In one line: a vector represents both a magnitude (size) and a direction — unlike a plain number, it tells you not just 'how much' but also 'which way,' making it perfect for describing motion and force.

Key Ideas

1Vector vs. Scalar. A scalar has only magnitude (like temperature or mass); a vector has both magnitude and direction (like velocity or force).

2Vector Notation. Often written as an arrow over a letter, or as component form (x,y), showing horizontal and vertical parts.

3Magnitude of a Vector. For a vector (x,y), magnitude = sqrt(x^2+y^2) — found using the Pythagorean theorem.

4Adding Vectors. Add corresponding components: (a,b) + (c,d) = (a+c, b+d).

5Scalar Multiplication of a Vector. Multiply each component by the scalar: k*(x,y) = (kx, ky), which scales the vector's length without changing its direction (unless k is negative, which reverses it).

Worked Examples

Find the magnitude of the vector (3,4).

Add the vectors (2,5) and (4,-3).

(6, 2)

Multiply the vector (3,-2) by the scalar 4.

(12, -8)

Formulas

Magnitude of (x,y) = sqrt(x^2+y^2)

(a,b) + (c,d) = (a+c, b+d)

k*(x,y) = (kx, ky)

Practice Yourself

Find the magnitude of (6,8).

Add (1,2) and (3,4).

(4,6)

Multiply (2,3) by scalar 5.

(10,15)