# Calculus 3 : Vectors and Vector Operations

## Example Questions

### Example Question #21 : Vectors And Vector Operations

Let  be any arbitrary real valued vector inclined at an angle  to the horizontal. Calculate the projection of the vector on the horizontal.

Explanation:

Projection means a shadow. If light is cast on the vector from above, it will cast a shadow on the horizontal plane.

We know

The hypotenuse here is the vector  itself. Solving for the adjacent side gives us the horizontal projection to be

### Example Question #22 : Vectors And Vector Operations

Find the angle between the vectors  and .

Explanation:

The formula for finding the angle between the vectors is the dot product formula, which is . First, we find the value of . Next we find the magnitude of both a and b.  and . Plugging in and solving for theta, we get . To get theta by itself, we take the inverse cosine of both sides.

### Example Question #23 : Vectors And Vector Operations

Find the angle between the vectors a and b, where , and

Explanation:

Using the formula for the cross product, which is , we have all the values except theta. Plugging in the known values and solving, we get . Therefore,

### Example Question #24 : Vectors And Vector Operations

The angle between a = (2, 1,1) and b = (1, 2,1) is:

None of the Above

Explanation:

In order to find the angle between two vectors we use:

so

and

so,

Therefore,

### Example Question #25 : Vectors And Vector Operations

Find the angle in degrees between the two vectors. Round the answer to the nearest tenth.

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 #26 : Vectors And Vector Operations

Find the angle between the two vectors.

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 #27 : Vectors And Vector Operations

Find the angle between vectors  and .

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 #28 : Vectors And Vector Operations

Find the angle between vectors a and b, where , and .

Explanation:

Using the formula for the dot product, which is

,

all we do not have is the value of theta.

Plugging in what we know, we get

.

Doing inverse cosine of both sides gets us

.

### Example Question #29 : Vectors And Vector Operations

Find the angle between the vectors a and b, where , and .

Explanation:

Using the formula for the cross product, which is

,

we have all the values except theta.

Plugging in the known values and solving, we get

.

Therefore,

### Example Question #30 : Vectors And Vector Operations

Find the angle between the vectors a and b, where , and .

Explanation:

Using the formula for the dot product, which is

,

we have all the values except theta.

Plugging in the known values and solving, we get

.

Therefore,