### All Calculus 3 Resources

## Example Questions

### Example Question #51 : Vectors And Vector Operations

Find the angle between the two vectors. Round to the nearest degree.

**Possible Answers:**

**Correct answer:**

In order to find the angle between the two vectors, we follow the formula

and solve for

Using the vectors in the problem, we get

Simplifying we get

To solve for

we find the

of both sides and get

and find that

### Example Question #51 : Vectors And Vector Operations

Find the angle between the two vectors. Round to the nearest degree.

**Possible Answers:**

**Correct answer:**

In order to find the angle between the two vectors, we follow the formula

and solve for

Using the vectors in the problem, we get

Simplifying we get

To solve for

we find the

of both sides and get

and find that

### Example Question #52 : Vectors And Vector Operations

Find the angle between the two vectors. Round to the nearest degree.

**Possible Answers:**

**Correct answer:**

In order to find the angle between the two vectors, we follow the formula

and solve for

Using the vectors in the problem, we get

Simplifying we get

To solve for

we find the

of both sides and get

and find that

### Example Question #53 : Vectors And Vector Operations

Calculate the angle between the vectors and , and express the measurement of the angle in degrees.

**Possible Answers:**

**Correct answer:**

The angle between the vectors and is given by the following equation:

where represents the cross product of the vectors and , and and represent the respective magnitudes of the vectors and .

We are given the vectors and . Calculate , , and , and then substitute these results into the formula for the angle between these vectors, as shown:

,

,

and

.

Hence,

The principal angle for which is . Hence, the angle between the vectors and measures .

### Example Question #55 : Vectors And Vector Operations

Find the angle between the gradient vector and the vector where is defined as:

**Possible Answers:**

**Correct answer:**

Find the angle between the gradient vector and the vector where is defined as:

_____________________________________________________________

**Compute the gradient** by taking the partial derivative for each direction:

At we have:

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The angle between two vectors and can be found using the dot product:

We wish to find the angle between the two vectors:

**Compute the dot product between and , **

**Therefore the dot product is: **

**Compute the magnitude of **

**Compute the magnitude of **** , **

**Now put it all together: **

### Example Question #1 : Distance Between Vectors

Given the vectors

find the distance between them, .

**Possible Answers:**

**Correct answer:**

To find the distance between the vectors

we do the following calculation:

### Example Question #1 : Distance Between Vectors

Find the distance between the two vectors

**Possible Answers:**

**Correct answer:**

To find the distance between the two vectors

we make the following calculation

### Example Question #1 : Distance Between Vectors

Find the Euclidian distance between the two vectors:

**Possible Answers:**

**Correct answer:**

The Euclidian distance between two vectors is:

Plugging in the numbers given, we have:

### Example Question #4 : Distance Between Vectors

Find the distance between if and .

**Possible Answers:**

**Correct answer:**

Write the formula to find the magnitude of the vector .

Substitute the points into the equation assuming and .

The answer is:

### Example Question #5 : Distance Between Vectors

Calculate the length of line segment *AB *given *A*(−5, −2, 0) and *B*(6, 0, 3):

**Possible Answers:**

None of the Above

**Correct answer:**

First we need to find

so

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