1. Forces in Two Dimensions
Chapter 5

2. Representing Vectors in One DimensionB A x -4 -3 -2 -1 1 2 3 4
Vector A has a magnitude of 3 units and is in the +x direction
Vector B has a magnitude of 2 units and is in the –x direction
We can also represent vectors as a symbol with an arrow over it—e.g., 𝐴 means the vector A.

3. Combining Vectors in One DimensionA B x -4 -3 -2 -1 1 2 3 4 5
Vector A has a magnitude of 3 units and is in the +x direction
Vector B has a magnitude of 2 units and is in the +x direction
If C = A + B, the resultant vector C has a magnitude of 5 units and is in the +x direction.
If D = A – B, the resultant vector D has a magnitude of 1 unit and is in the +x direction (because A – B = A + (-B) and the vector –B has the same magnitude as B and points in the opposite direction).
We can also multiply vectors by scalars. The vector E = nA is a vector whose magnitude is n times the magnitude of A and points in the same direction as A. So if n=3 then the magnitude of E is 9 units and it is in the +x direction.

4. Combining Vectors in Two Dimensions
Add vectors by placing them tip to tail. The resultant vector is the vector connecting the tail of the first vector to the tip of the second vector. So for 𝑅 = 𝐴 + 𝐵 :
If the angle 𝜃 between 𝐴 and 𝐵 is a right angle:
𝑅 2 = 𝐴 2 + 𝐵 2 𝑅 𝐵
Pythagorean Theorem
Otherwise:
𝑅 2 = 𝐴 2 + 𝐵 2 −2𝐴𝐵 cos 𝜃 𝐴
The Law of Cosines

5. Resolving Components of Vectors+y
𝐴 = 𝐴 𝑥 + 𝐴 𝑦 𝐴
𝐴 𝑦
𝐴 𝑥 =𝐴 cos 𝜃
𝐴 𝑦 =𝐴 sin 𝜃 𝜃 +x
𝐴 𝑥

6. Adding Vectors Using Components
Once we’ve resolved vectors into their x- and y-components, we can just add the components and determine the resultant vector. +y
𝑅 𝑥 = 𝐴 𝑥 + 𝐵 𝑥
𝑅 𝑦 = 𝐴 𝑦 + 𝐵 𝑦 𝑅 𝐵
𝐵 𝑦 𝐵
𝜃= tan −1 𝑅 𝑦 𝑅 𝑥
𝐴 𝑦 𝐴 𝐴 +x
𝐴 𝑥
𝐵 𝑥

7. Activities with Vectors
Name That Vector Interactive
Vector Addition Interactive