Vector Addition
Helena Dedic
Ex 1
Three vectors are specified as follows: A is 5 m at
N of E, B is 7 m at
E of S, and C is 4 m at
W of S. Find the magnitude and direction of their sum.
Solution
We start by drawing a diagram that shows each vector with its tail at the origin and then by computing the angle with x-axis.
![Vectors Ex first Sol.png](/gwikis/pwiki/images/5/5f/Vectors_Ex_first_Sol.png)
We calculate the value of each component.
We then compute the magnitude and the direction of R:
Ex 2
Four vectors, each of magnitude 2 m, are shown in the figure.
a. Express each in unit vector notation.
b. Express their sum in unit vector notation.
c. What is the magnitude and direction of their sum?
![Vectors Ex1.png](/gwikis/pwiki/images/d/d9/Vectors_Ex1.png)
Solution
a) We start by placing the tail of each vector at the origin of rectangular coordinates and finding the x- and y-components of each vector:
m
m
m
m
![Vectors Ex1 Sol.png](/gwikis/pwiki/images/0/0f/Vectors_Ex1_Sol.png)
b) We can predict the sum A+ B + C + D by doing a rough sketch, adding the vectors graphically. C and D roughly cancel and we can predict that the sum of A and B is in the fourth quadrant.
![Vectors Ex1 Sol b.png](/gwikis/pwiki/images/c/c8/Vectors_Ex1_Sol_b.png)
We add the vectors by adding x- and y-components
The sum, as predicted, is in the fourth quadrant.
c) We find that the vector sum has the magnitude 0.73 m and it points
below the x-axis.
Ex 3
The vector A has a magnitude of 6 m and vector B has a magnitude of 4 m. What is the angle between them if the magnitude of their resultant is:
a. the maximum possible
b. the minimum possible
c. 3 m
d. 8 m
Do each part graphically and by components. (Let A lie along the x-axis.)
Solution
![Vectors Ex2 Sol.png](/gwikis/pwiki/images/9/99/Vectors_Ex2_Sol.png)
![Vectors Ex2 Sol b.png](/gwikis/pwiki/images/0/0f/Vectors_Ex2_Sol_b.png)
Ex 4
The displacement A is 6 m east. Find displacement B if the magnitude of A - B is half that of A and points in the direction
N of E.
Solution
We define C = A - B
Solving for B:
B = A - C
We know that C = 3 m and its directions as measured from the positive x-axis is
. We can compute the components of C as follows:
Given that
and
we compute the components of B