# High School Math : Parametric, Polar, and Vector

## Example Questions

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### Example Question #5 : Understanding Vector Calculations

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 #6 : Understanding Vector Calculations

Find the unit vector of .

Explanation:

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

unit vector:

### Example Question #7 : Understanding Vector Calculations

Is a unit vector?

yes, because magnitude is equal to

not enough information given

no, because magnitude is not equal to

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 #8 : 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 #9 : 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 #10 : Understanding Vector Calculations

Given vector and , solve for .

Explanation:

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:

### Example Question #21 : Calculus Ii — Integrals

Find the magnitude of .

Explanation:

therefore the vector is

To solve for the magnitude:

### Example Question #22 : Calculus Ii — Integrals

Let  and  be the following vectors:   and . If  is the acute angle between the vectors, then which of the following is equal to ?

Explanation:

The cosine of the acute angle between two vectors is given by the following formula:

, where  represents the dot product of the two vectors,   is the magnitude of vector a, and  is the magnitude of vector b.

First, we will need to compute the dot product of the two vectors. Let's say we have two general vectors in space (three dimensions),  and . Let the components of  be  and the components of  be . Then the dot product  is defined as follows:

.

Going back to the original problem, we can use this definition to find the dot product of   and .

The next two things we will need to compute are  and

Let the components of a general vector  be . Then  is defined as .

Thus, if   and , then

and

.

Now, we put all of this information together to find the cosine of the angle between the two vectors.

We just need to simplify this.

.

In order to get it completely simplified, we have to rationalize the denominator by multiplying the numerator and denominator by the sqare root of 21.

.

We just have one more step. We need to solve for the value of the angle. In order to do this, we can take the inverse cosine of both sides of the equation.

.