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Week 2
#notes
Vector Resultants and Components
F = F
1
^ e 1
+ F
2
e ^ 2
^ v → are unit vectors which give u the direction of the vector
You have to decompose vectors into its x and y components to get the magnitude of the vector.
( keep in mind the directions of the vectors Week 1)
By taking this into account the above equation can be simplified into:
F = F cos ( θ )^ e 1
When you have a vector given that is described by the slope, you are able to understand the
direction from just the slope.
To calculate the force now we could use the cosine and sine functions
F = F (4/5)^ e 1
cos( θ ) = 4/5, sin( θ ) = 3/
Systems of vectors
Find
F
R
To find the resultant force we must break each vector into its components and then add/
subtract them in the respective directions to end up with the resultant force.
F 1 = 600(cos(60°))^ i + 600(sin(60°))^ j
F 2 = −300(12/13)
^
i + 300(5/13)
^
j
F
3
= −1000(cos(30°))^ i + 1000(sin(30°))^ j
FR =
F 1 +
F 2 +
F 3
FR = 523
^
i − 231
^
j
To find the magnitude:
F
R
2 ) + (−
2 ))
Vector Resultants and Components (3d)
We will expressing 3d vectors as the sum of three orthogonal components
P = P
x
^
i + P y
^
j + P z
^
k
P | = √ P
2
x
+ P
2
y
+ P
2
z
Instead of having one angle to deal with while looking at 3d vectors we have three angles to
deal with.
Degrees of freedom
Translate in x, y and z axis
Rotation about its axis
So it has 6 degrees of freedom
Finding the direction of the vector:
Sweep through space to get from positive axis to the vector
Example Problem
Given that:
F
1
= 300^ i + 200^ j − 600^ kN
F
2
= 400 N α = 45° β = 120°
Find:
FR | and the Direction of the vector
Solution:
By using Euler's rule ^7f6de9 we are able to find out γ
cos 45°
2
2
2
= 1
γ = 60°
We are only given the magnitude of
F
2
so we need to find the magnitude
F
2
= 400 cos 45° + 400 cos 120° + 400 cos 60°
The resultant force would be the sum of Force 1 and Force 2
FR = 582.
^
i + 0
^
j − 400
^
k
F
R
| = 707 N
To find the direction
cos α r
cos γ r
Position Vectors
Three arbitrary points in space can be measured relative to each other by subtracting the
distances between them in each axis.
Tension always pulls on things
