### All Precalculus Resources

## Example Questions

### Example Question #1 : Find The Measure Of An Angle Between Two Vectors

Find the angle between the following two vectors in 3D space.

**Possible Answers:**

**Correct answer:**

We can relate the dot product, length of two vectors, and angle between them by the following formula:

So the dot product of

and

is the addition of each product of components:

now the length of the vectors of a and b can be found using the formula for vector magnitude:

So:

hence

### Example Question #1 : Find The Measure Of An Angle Between Two Vectors

The dot product may be used to determine the angle between two vectors.

Use the dot product to determine if the angle between the two vectors.

,

**Possible Answers:**

**Correct answer:**

First, we note that the dot product of two vectors is defined to be;

.

First, we find the left side of the dot product:

.

Then we compute the lengths of the vectors:

.

We can then solve the dot product formula for theta to get:

Substituting the values for the dot product and the lengths will give the correct answer.

### Example Question #1 : Find The Measure Of An Angle Between Two Vectors

Find the angle between the two vectors:

**Possible Answers:**

**Correct answer:**

Solving the dot product formula for the angle between the two vectors results in the equation .

If we call the vectors a and b, finding the dot product and the lengths of the vectors, then substituting them into the formula will give the correct angle.

Substituting the values correctly will give the correct answer.

### Example Question #3 : Find The Measure Of An Angle Between Two Vectors

Find the measure of the angle between the following vectors:

**Possible Answers:**

**Correct answer:**

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

### Example Question #4 : Find The Measure Of An Angle Between Two Vectors

Find the measure of the angle between the two vectors.

**Possible Answers:**

**Correct answer:**

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.

### Example Question #5 : Find The Measure Of An Angle Between Two Vectors

Find the angle between the vectors and .

**Possible Answers:**

**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

### Example Question #6 : Find The Measure Of An Angle Between Two Vectors

Find the angle between the vectors and . Note that the first vector is in polar form and the second is in component form.

**Possible Answers:**

**Correct answer:**

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

### Example Question #7 : Find The Measure Of An Angle Between Two Vectors

Find where and . Note that is in component form while is in polar form.

**Possible Answers:**

**Correct answer:**

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

### Example Question #1 : Find The Dot Product Of Two Vectors

Find the dot product of the two vectors

and

.

**Possible Answers:**

**Correct answer:**

To find the dot product of two vectors,

find the product of the x-components,

find the product of the y-components,

and then find the sum.

### Example Question #1 : Find The Dot Product Of Two Vectors

Find the dot product of the two vectors

and

.

**Possible Answers:**

**Correct answer:**

To find the dot product of two vectors,

find the product of the x-components,

find the product of the y-components,

and then find the sum.