Exercises on Vectors

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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.


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

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


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:


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

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.)


Vectors Ex2 Sol.png
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.


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