Evaluate geometric vectors - Pre-Calculus

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.

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: .

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.

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.

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:

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.

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:

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.

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:

Finding the resultant requires us to add like components:

Finding the resultant requires us to add like components:

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:

We find the norm of a vector by finding the sum of each element squared and then taking the square root.

.

When multiplying a vector by a scalar we multiply each component of the vector by the scalar and the result is a vector:

We find the norm of a vector by finding the sum of each component squared and then taking the square root of that sum.

In order to simplify this problem we need to multiply the scalar factor to each component of the vector.

In our case the scalar factor is

Thus,

The norm of a vector is also known as the length of the vector. The norm is given by the formula:

.

Here, we have

,

the correct answer.

Given a scalar k and a vector v, the vector given by their products is defined component-wise:

.

Here, our product is:

Write the formula to find the norm, or the length the vector.

Substitute the known values of the vector and solve.