# Precalculus : Parallel and Perpendicular Vectors in Two Dimensions

## Example Questions

### Example Question #11 : Determine If Two Vectors Are Parallel Or Perpendicular

Which of the following pairs of vectors are parallel?

Explanation:

For two vectors,  and  to be parallel, , for some real number .

Recall that for a vector, .

This is the correct answer because .

### Example Question #12 : Determine If Two Vectors Are Parallel Or Perpendicular

Which of the following pairs of vectors are parallel?

Explanation:

For two vectors,  and  to be parallel, , for some real number .

Recall that for a vector, .

.

Using the formula  we have our  to be .

Applying this we find the vector that is parallel.

### Example Question #11 : Determine If Two Vectors Are Parallel Or Perpendicular

Which pair of vectors represents two parallel vectors?

Explanation:

Two vectors are parallel if their cross product is . This is the same thing as saying that the matrix consisting of both vectors has determinant zero.

This is only true for the correct answer.

In essence each vector is a scalar multiple of the other.

### Example Question #14 : Determine If Two Vectors Are Parallel Or Perpendicular

Which relationship best describes the vectors and ?

perpendicular

the same direction, but different magnitudes

neither parallel nor perpendicular

parallel

perpendicular

Explanation:

We can discover that these vectors are perpendicular by finding the dot product:

A dot product of zero for two non zero vectors means that they are perpedicular vectors.

### Example Question #15 : Determine If Two Vectors Are Parallel Or Perpendicular

Which relationship best describes the two vectors and ?

perpendicular

neither parallel nor perpendicular

parallel

both parallel and perpendicular

parallel

Explanation:

To show that these are parallel, we have to find their magnitudes using the Pythagorean Theorem:

To multiply , multiply the numbers within the radicals first, then take the square root:

Now we have to find the dot product and compare it to the product of these two magnitudes. If they are the same, or if they differ only by sign [one is the negative version of the other] then the two lines are parallel.

This is the negative version of the magnitudes' product! This means that the two vectors are parallel.

### Example Question #16 : Determine If Two Vectors Are Parallel Or Perpendicular

Find the dot product of the two vectors

and

.

Explanation:

To find the dot product of two vectors,

find the product of the x-components,

find the product of the y-components,

and then find the sum.

### Example Question #17 : Determine If Two Vectors Are Parallel Or Perpendicular

Find the dot product of the two vectors

and

.

Explanation:

To find the dot product of two vectors,

find the product of the x-components,

find the product of the y-components,

and then find the sum.

### Example Question #18 : Determine If Two Vectors Are Parallel Or Perpendicular

Find the dot product of the two vectors

and

.

Explanation:

To find the dot product of two vectors,

find the product of the x-components,

find the product of the y-components,

and then find the sum.

### Example Question #19 : Determine If Two Vectors Are Parallel Or Perpendicular

Evaluate the dot product of the following two vectors:

Explanation:

To find the dot product of two vectors, we multiply the corresponding terms of each vector and then add the results together, as expressed by the following formula:

### Example Question #20 : Determine If Two Vectors Are Parallel Or Perpendicular

Let

Find the dot product of the two vectors

.