# Exercises on Vectors

### 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.

We calculate the value of each component.

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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?

**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

**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.

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**

#### 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**

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