To find the angle between two vectors, use the following formula:

 is known as the dot product of two vectors. It is found via the following formula:

The denominator of the fraction involves multiplying the magnitude of each vector. To find the magnitude of a vector, use the following formula:

Now we have everything we need to find our answer. Use our given vectors:

So the angle between these two vectors is 102.09 degrees

We use the dot product to find the angle between two vectors. The dot product has two formulas:

We solve for the angle measure to find the computational formula:

Our vectors give dot products and lengths:

Substituting these values into the formula above will give the correct answer.

To determine the angle between our two vectors, we can use the fact that for any 2 vectors and , where is the magnitude and is the angle between the 2 vectors, which is what we are looking for.

Working from the left, we can first find the dot product,

Now we'll find the magnitudes of the two vectors by using the Pythagorean Theorem:

take the square root of both sides

Now we can plug these values back into the equation to start solving for theta:

multiply the two numbers inside the radicals:

divide both sides by

 take the inverse cosine of both sides

To find the angle between two vectors, we can use the fact that . In order to find the dot product, we need to convert the vector to component form. This is easiest to do after drawing a quick sketch of the vector:

To find the vertical component, set up an equation involving sine, since the vertical component is the side of a right triangle across from the 20-degree angle:

evaluate the sine of 20, then multiply by 13

To find the horizontal component, set up an equation involving cosine, since the horizontal component is the side of a right triangle adjacent to the 20-degree angle:

evaluate the cosine of 20, then multiply by 13

The component form of this vector can be written as

Next, we can find the magnitude of the vector already in component form by using the Pythagorean Theorem:

take the square root of both sides

Now we have all the information that we need to solve for theta.

Find the dot product for the left side:

which we can now put back in the equation:

now divide both sides by

take invese cosine of both sides

First, convert to component form. The easiest way to do that would be with special right triangles, knowing that for a 45-45-90 triangle the ratio of the hypotenuse to both of the legs is . So if the magnitude of the vector is , then the components must both be 3.

This could also be found by setting up and solving the equations:

and

Now that both vectors are component form, we see that we're multiplying

Two vectors are perpendicular if their dot product is zero, and parallel if their dot product is 1.

Take the dot product of our two vectors to find the answer:

Using our given vectors:

Thus our two vectors are perpendicular.

When we multiply a scalar (regular number) by a matrix, all we need to do is mulitply it to every entry inside the matrix: 

When we multiply a scalar (regular number) by a matrix, all we need to do is mulitply it through to every entry inside the matrix:

When we multiply a scalar (regular number) by a matrix, all we need to do is mulitply it through to every entry inside the matrix:

We note k is simply a scalar. To do this multiplication all we need to do is to multply each entry of the matrix by k.

we see that when we multiply we have :

this gives the entry of the matrix kA.

Therefore the resulting matrix is :