### All High School Math Resources

## Example Questions

### Example Question #1 : Understanding Vector Calculations

Let be vectors. All of the following are defined EXCEPT:

**Possible Answers:**

**Correct answer:**

The cross product of two vectors (represented by "x") requires two vectors and results in another vector. By contrast, the dot product (represented by "") between two vectors requires two vectors and results in a scalar, not a vector.

If we were to evaluate , we would first have to evaluate , which would result in a scalar, because it is a dot product.

However, once we have a scalar value, we cannot calculate a cross product with another vector, because a cross product requires two vectors. For example, we cannot find the cross product between 4 and the vector <1, 2, 3>; the cross product is only defined for two vectors, not scalars.

The answer is .

### Example Question #1 : Understanding Vector Calculations

Find the magnitude of vector :

**Possible Answers:**

**Correct answer:**

To solve for the magnitude of a vector, we use the following formula:

### Example Question #1 : Understanding Vector Calculations

Given vector and , solve for .

**Possible Answers:**

**Correct answer:**

To solve for , we need to add the components in the vector and the components together:

### Example Question #1 : Understanding Vector Calculations

Given vector and , solve for .

**Possible Answers:**

**Correct answer:**

To solve for , we need to subtract the components in the vector and the components together:

### Example Question #3 : Understanding Vector Calculations

Given vector and , solve for .

**Possible Answers:**

**Correct answer:**

To solve for , We need to first multiply into vector to find and multiply into vector to find ; then we need to subtract the components in the vector and the components together:

### Example Question #2 : Understanding Vector Calculations

Find the unit vector of .

**Possible Answers:**

**Correct answer:**

To solve for the unit vector, the following formula must be used:

unit vector:

### Example Question #11 : Calculus Ii — Integrals

Is a unit vector?

**Possible Answers:**

no, because magnitude is not equal to

not enough information given

yes, because magnitude is equal to

**Correct answer:**

yes, because magnitude is equal to

To verify where a vector is a unit vector, we must solve for its magnitude. If the magnitude is equal to 1 then the vector is a unit vector:

** is a unit vector because magnitude is equal to .**

### Example Question #2 : Understanding Vector Calculations

Given vector . Solve for the direction (angle) of the vector:

**Possible Answers:**

**Correct answer:**

To solve for the direction of a vector, we use the following formula:

=

with the vector being

### Example Question #11 : Calculus Ii — Integrals

Solve for vector given direction of and magnitude of .

**Possible Answers:**

**Correct answer:**

To solve for a vector with the magnitude and direction given, we use the following formula:

### Example Question #12 : Calculus Ii — Integrals

Given vector and , solve for .

**Possible Answers:**

**Correct answer:**

To solve for , We need to multiply into vector to find ; then we need to subtract the components in the vector and the components together: