# High School Math : Understanding Vector Calculations

## Example Questions

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### Example Question #1 : Understanding Vector Calculations

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

Explanation:

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.

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

Find the magnitude of vector :

Explanation:

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

### Example Question #1 : Understanding Vector Calculations

Given vector and , solve for .

Explanation:

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 .

Explanation:

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

### Example Question #2 : Vector

Given vector and , solve for .

Explanation:

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 #11 : Calculus Ii — Integrals

Find the unit vector of .

Explanation:

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

unit vector:

### Example Question #1 : Understanding Vector Calculations

Is a unit vector?

no, because magnitude is not equal to

yes, because magnitude is equal to

not enough information given

yes, because magnitude is equal to

Explanation:

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 #1 : Understanding Vector Calculations

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

Explanation:

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

=

with the vector being

### Example Question #1 : Understanding Vector Calculations

Solve for vector  given direction of  and magnitude of .

Explanation:

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

### Example Question #1 : Understanding Vector Calculations

Given vector and , solve for .