# Precalculus : Geometric Vectors

## Example Questions

### Example Question #55 : High School Physics

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 #56 : High School Physics

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 #57 : High School Physics

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 #58 : High School Physics

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 #59 : High School Physics

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 #60 : High School Physics

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 #61 : High School Physics

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.

### Example Question #62 : High School Physics

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

Find  using the parallelogram 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 :

Place the tails of  and  at the same point:

Construct a parallelogram:

Construct and measure the resultant using ruler and protractor:

### Example Question #11 : Geometric Vectors

Find .

Explanation:

Finding the resultant requires us to add like components:

Find .