• The magnitude of a vector is represented by its length.
• You can multiply the magnitude of a vector by a scalar quantity to change its length.
A = 10m
0.5A =5m
2A =20m

• All vectors must have a direction.
– North, south, east, west
– Or an angle
• Sometimes measured from the horizontal or vertical
• Sometimes measured from the positive x-axis and around to 360 o
230 o
30 o
40 o

• Must manipulate all vectors so they are put “tip to tail”
30 o
A = 10m
40 o
• Resultant points from start to finish.

• Must manipulate all vectors so they are put “tip to tail”
• Resultant points from start to finish.
40 o
A = 10m

• Must manipulate all vectors so they are put “tip to tail”
• Resultant points from start to finish.
30 o
A = 10m
40 o

• Flip the direction of the one being subtracted
• Then put tip to tail and follow adding procedure
A = 10m
40 o

• Every vector will have a…
– Horizontal component that points directly left or right
– Vertical component that points directly up or down.
• The components should be drawn tip to tail and lead to the same point as the original vector.
• The angle goes by the start.
𝜽
Horizontal component

• In this case…
– Horizontal component is the adjacent side which can be calculated using 𝐴 = 𝐴 cos 𝜃 = 25m cos40 o = 19m 𝑥
– Vertical component is the opposite side which can be calculated using 𝐴 𝑦
= 𝐴 𝑠𝑖𝑛 𝜃 = 25m sin 40 o = 16m
• REMEMBER: horizontal is not always cosine. It depends where the angle is located!
– If the angle is to the vertical, sine and cosine would flip.
𝜽 = 𝟒𝟎 𝒐
Horizontal component

Find the components of the following vectors
A = 10m
𝑨 𝒙
𝑨 𝒚
= +𝟏𝟎𝒎
= 𝟎 m
30 o
40 o
𝐴 𝑥
= 𝐴 cos 𝜃
= 15m cos40 o
= 11.5m
𝐴 𝑦
= 𝐴 𝑠𝑖𝑛 𝜃
= 15m sin 40 o
= 9.6m
𝐴 𝑥
= 𝐴 sin 𝜃
= 20m sin 30 o
= 10.0 m
𝐴 𝑦
= 𝐴 𝑐𝑜𝑠 𝜃
= 20m cos 30 o
= 17.3m

• Step 1: find the components of each vector
• Step 2: add all of the x-components together to find the resultant’s xcomponent
• Step 3: add all of the y-components together to find the resultant’s ycomponent
• Step 4: build your actual resultant out of its components you just fund.
• Step 5: use Pythagorean Theorem and SohCahToa to find magnitude and direction of resultant.
A = 10m
𝑨 𝒙
𝑨 𝒚
= +𝟏𝟎𝒎
= 𝟎 m 𝑎 2
12.5
2
+ 𝑏 2
+ 9.6
= 𝑐
2
2
= 𝑐 2 c= 15.8m
tan 𝜃 =
𝐴 𝑦
= 𝐴 𝑠𝑖𝑛 𝜃
= 15m sin 40 o
RESULTANT
X-component:
10m+11.5m = 21.5m
Y- component:
0m+9.6m = 9.6m
𝐴 𝑥
= 21.5𝑚
𝐴 𝑦
= 9.6m
40 o
𝐴 𝑥 tan 𝜃 =
= 𝐴 cos 𝜃
12.5
𝜽 = 𝟑𝟕. 𝟓 𝒐
= 15m cos40 o
= 11.5m