Adding Vectors that are
not perpendicular
Holt Physics
Chapter 3 Section 2 (continued)
Non-perpendicular vectors
 Because vectors do not always form right
triangles, you cannot automatically apply the
Pythagorean theorem and tangent function to
the original vectors
Vector 2
Resultant vector
Vector 1
No right triangle!
Non-perpendicular vectors
 This will require new “old” variables
For Displacement Vectors
Vector #1
Δx1
Δy1
DR1
θ1
Vector #2
Δx2
Δy2
DR2
θ2
Resultant
ΔxT
ΔyT
DRT
θR
Non-perpendicular vectors
 Step #1
You must resolve vector #1 into x & y components
Vector 1
DR1
Δy1
θ1
Δx1
Non-perpendicular vectors
 Step #2
You must resolve vector #2 into x & y components
DR2
Δy2
Vector 2
Δx2
θ2
Non-perpendicular vectors
 Step #3
Add all X components to find ΔxT
Δx1 + Δx2 = ΔxT
Δx1
Δx2
ΔxT
Non-perpendicular vectors
 Step #4
Add all Y components to find ΔyT
Δy1 + Δy2 = ΔyT
Δy2
ΔyT
Δy1
Non-perpendicular vectors
 Step #5
Now you have total x & y components
Use Pythagorean theorem to find resultant
DRT2 = ΔxT2 + ΔyT2
DRT
ΔyT
θR
ΔxT
Non-perpendicular vectors
 Step #6
Use tangent to find the angle
- Same equations -
Tan θR = ΔyT
ΔxT
 θR = Tan-1 (ΔyT / ΔxT)
Non-perpendicular vectors
You may wish to draw the x-total and y-total
vectors into the original drawing if it helps
you, or make a new triangle with just the
totals.
Vector 2
Resultant vector
DRT
Vector 1
θR
ΔyT
ΔxT
Non-perpendicular vectors
* If you have velocities instead of displacement:
 Replace “Δ” and “D” with “V”
* If you have more than two vectors, the third
vector’s variables will be “Δx3, ….”, and so on.