### All Calculus 3 Resources

## 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.

**Possible Answers:**

**Correct answer:**

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 .

**Possible Answers:**

**Correct answer:**

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

**Possible Answers:**

**Correct answer:**

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:

**Possible Answers:**

None of the Above

**Correct answer:**

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.

**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 #26 : Vectors And Vector Operations

Find the angle between the two vectors.

**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 #27 : Vectors And Vector Operations

Find the angle between vectors and .

Round your answer 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 #28 : Vectors And Vector Operations

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

**Possible Answers:**

**Correct answer:**

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 .

**Possible Answers:**

**Correct answer:**

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 .

**Possible Answers:**

**Correct answer:**

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,