# Precalculus : Find the Sum and Difference of Vectors

## Example Questions

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### Example Question #1 : Find The Sum And Difference Of Vectors

Evaluate

Explanation:

When adding two vectors, they need to be expanded into their components. Luckily, the problem statement gives us the vectors already in their component form. From here, we just need to remember that we can only add like components. So for this problem we get:

Now we can combine those values to write out the complete vector:

### Example Question #2 : Find The Sum And Difference Of Vectors

What is the magnitude and angle for the following vector, measured CCW from the x-axis?

Explanation:

The magnitude of the vector is found using the distance formula:

To calculate the angle we must first find the inverse tangent of :

This angle value is the principal arctan, but it is in the fourth quadrant while our vector is in the second. We must add the angle 180° to this value to arrive at our final answer.

### Example Question #3 : Find The Sum And Difference Of Vectors

Vector has a magnitude of 3.61 and a direction 124° CCW from the x-axis. Express  in unit vector form.

Explanation:

For vector , the magnitude is doubled, but the direction remains the same.

For our calculation, we use a magnitude of:

The x-coordinate is the magnitude times the cosine of the angle, while the y-coordinate is the magnitude times the sine of the angle.

The resultant vector is: .

### Example Question #31 : Understanding Scalar And Vector Quantities

What are the magnitude and angle, CCW from the x-axis, of ?

Explanation:

When multiplying a vector by a constant (called scalar multiplication), we multiply each component by the constant.

The magnitude of this new vector is found with these new components:

To calculate the angle we must first find the inverse tangent of :

This is the principal arctan, but it is in the first quadrant while our vector is in the third. We to add the angle 180° to this value to arrive at our final answer.

### Example Question #32 : Understanding Scalar And Vector Quantities

Vector has a magnitude of 2.24 and is at an angle of 63.4° CCW from the x-axis. Vector has a magnitude of 3.16 at an angle of 342° CCW from the x-axis.

Find  by using the nose-to-tail graphical method.

Explanation:

First, construct the two vectors using ruler and protractor:

Place the tail of  at the nose of :

Construct the resultant  from the tail of to the nose of :

With our ruler and protractor, we find that is 4.12 at an angle of 14.0° CCW from the x-axis.

### Example Question #6 : Geometric Vectors

Find the magnitude and angle CCW from the x-axis of  using the nose-to-tail graphical method.

Explanation:

Construct and from their x- and y-components:

Since we are subtracting, reverse the direction of :

Form  by placing the tail of  at the nose of :

Construct and measure the resultant, , from the tail of to the nose of  using a ruler and protractor:

### Example Question #33 : Understanding Scalar And Vector Quantities

Express a vector with magnitude 2.24 directed 63.4° CCW from the x-axis in unit vector form.

Explanation:

The x-coordinate is the magnitude times the cosine of the angle, while the y-coordinate is the magnitude times the sine of the angle.

The resultant vector is: .

### Example Question #34 : Understanding Scalar And Vector Quantities

Vector has a magnitude of 2.24 and is at an angle of 63.4° CCW from the x-axis. Vector has a magnitude of 3.61 and is at an angle of 124° CCW from the x-axis.

Find  by using the nose-to-tail graphical method.

Explanation:

First, construct the two vectors using ruler and protractor:

is twice the length of , but in the same direction:

Since we are subtracting, reverse the direction of :

Form  by placing the tail of  at the nose of :

Construct and measure the resultant  from the tail of to the nose of  with a ruler and protractor.

### Example Question #35 : Understanding Scalar And Vector Quantities

Vector has a magnitude of 2.24 and is at an angle of 63.4° CCW from the x-axis. Vector has a magnitude of 3.16 at an anlge of 342° CCW from the x-axis.

Find  by using the parallelogram graphical method.

Explanation:

First, construct the two vectors using ruler and protractor:

Place the tails of both vectors at the same point:

Construct a parallelogram:

Construct and measure the resultant using ruler and protractor:

### Example Question #36 : Understanding Scalar And Vector Quantities

Find  using the parallelogram graphical method.

Explanation:

Construct and from their x- and y-components:

Since we are subtracting, reverse the direction of :

Place the tails of and  at the same point:

Construct a parallelogram:

Construct and measure the resultant using a ruler and protractor.

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