# Calculus 3 : Vectors and Vector Operations

## Example Questions

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

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

Explanation:

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.

Explanation:

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.

Explanation:

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.

Explanation:

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:

Explanation:

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

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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, .

Explanation:

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

Explanation:

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:

Explanation:

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 .

Explanation:

Write the formula to find the magnitude of the vector .

Substitute the points into the equation assuming  and .

### Example Question #5 : Distance Between Vectors

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