In this explainer, we will learn how to recognize parallel and perpendicular vectors in 2D.

Let us begin by considering parallel vectors. Two vectors are parallel if they are scalar multiples of one another. In the diagram below, vectors , , and are all parallel to vector and parallel to each other.

We define parallel vectors in the following way.

### Definition: Parallel Vectors

Vectors and are parallel if for any scalar , where .

Note that if we have parallel vectors and such that , where is a positive value, then vectors and are parallel and point in the same direction. If is negative, then vectors and are parallel but point in opposite directions. In the figure above, vectors , , and are parallel in the same direction, but the parallel vector points in the opposite direction to that of the others.

We can see how we can apply this information in the first example.

### Example 1: Identifying Whether the Given Vectors Are Parallel

True or False: Vectors and are parallel.

- True
- False

### Answer

Two vectors and are parallel if they are scalar multiples of one another, that is, for any scalar , where .

Therefore, we check if there is a constant for which

Given that and , we have

Evaluating the -components gives

Taking the -components, we have

Thus, we could write that and the statement that vectors and are parallel is true.

Next, we consider how to identify perpendicular vectors. We can recall that to calculate the dot product of two vectors, we write them in component form, multiply the corresponding components of each vector, and add the resulting numbers.

### Definition: Dot Product of Two-Dimensional Vectors

The dot product of two vectors and is given by

Additionally, we can write that where is the angle between and .

We can use the angle definition of the dot product to help us identify perpendicular vectors. If the two vectors and are perpendicular, then the angle .

Thus, we can write

We can use this fact to identify that if the dot product of a given pair of vectors is equal to 0, then the vectors must be perpendicular. It is common to see the angle definition of the dot product used to find the angle between two given vectors. As an aside, if the angle or , then the vectors are parallel.

### Definition: Perpendicular Vectors

Two vectors and are perpendicular if

Equivalently, by calculating the dot product, and are perpendicular if

In the next example, we can see how we can apply this rule to identify a pair of perpendicular vectors.

### Example 2: Identifying a Pair of Perpendicular Vectors

Which of the following vector pairs are perpendicular?

### Answer

We recall that two vectors, and , are perpendicular if

To calculate , we multiply the corresponding components of each vector and add the resulting numbers:

Therefore, for each pair of vectors above, we check if

For the first vector pair, and , we have

As , and are not perpendicular.

The dot product of the second pair of vectors in option B, and , can be found to be

As , and are not perpendicular.

Next, the dot product of the vectors and is

Therefore, and are not perpendicular.

The final pair of vectors in option D, and , have a dot product of

As the dot product is equal to zero, and are perpendicular.

In the next example, we will see how we can find an unknown value in a pair of vectors, given that the vectors are parallel.

### Example 3: Finding the Value of the Constant to Make Two Vectors Parallel

If and , then one of the values of that makes is .

- 7
- 5

### Answer

We recall that two vectors and are parallel, , if one is a scalar multiple of the other.

We can write this as for any scalar , where .

We can substitute the values and to find the value of , giving

Evaluating the -components gives

Dividing through by , we have

We evaluate the -components above to give

Substituting the value into this equation, we have

Expanding the parentheses and subtracting from both sides of the equation give

Simplifying this equation by adding to both sides and then multiplying by 3, we have

Therefore, one of the values of that makes is the answer given in option A, which is 7.

We can check our answer by substituting into vectors and and checking if . This gives

We can see that our vectors are scalar multiples of each other. In this case, .

We check the other answer options. In option B, . So if , then we would have

However, there is no value of that would make this valid, so and when .

In answer option C, . If , then we would have

However, there is no value of that would be valid, so and when .

Finally, checking answer option D by substituting into gives

However, there is no valid value of here, so and when .

Therefore, we have shown that the only valid answer from the available options here is .

In the next example, we will see how we can identify an unknown value in a vector, given that the vectors are perpendicular.

### Example 4: Finding the Value of the Constant to Make Two Vectors Perpendicular

If , , and , then .

- 2
- 4

### Answer

We can recall that if two vectors and are perpendicular, , then their dot product is 0.

This means that for any perpendicular vectors and ,

We are given that and . Therefore,

Since , we can write that

Simplifying by adding 8 to both sides and then dividing through by 2 gives

We can then take the square root of both sides of the equation, but we must consider the positive and negative values of this. Thus,

Therefore, we can give the answer that the values of must be those given in option C: .

In the final example, we will identify if a given pair of vectors are parallel, perpendicular, or neither.

### Example 5: Identifying Whether the Given Vectors Are Parallel, Perpendicular, or Neither

Fill in the blank: Vectors and are .

### Answer

We may approach this question by considering if vectors and are parallel, perpendicular, or neither.

We can recall that two vectors are parallel if they are scalar multiples of one another. Here, we can check if for any scalar , where .

Substituting the given vectors, and , we check if there is a value of for which

Evaluating the -components gives

Evaluating the -components gives

However, as both values of are different, there is no single value of that solves the equation. So,

Therefore, vectors and are not parallel.

Next, we can check if and are perpendicular by finding their dot product.

The dot product of two vectors and is given by

Therefore, the dot product of and is given by

We recall that two vectors are perpendicular if their dot product is 0.

Therefore, we can answer the question: vectors and are perpendicular.

We now summarize the key points.

### Key Points

- Vectors and are parallel if for any scalar , where .
- Vectors and are perpendicular if their dot product is 0: