Lesson Explainer: Parallel and Perpendicular Vectors in 2D Mathematics

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 𝑎, 𝑏, and 𝑐 are all parallel to vector 𝑢 and parallel to each other.

We define parallel vectors in the following way.

Definition: Parallel Vectors

Vectors 𝑢 and 𝑣 are parallel if 𝑢=𝑘𝑣 for any scalar 𝑘, where 𝑘0.

Note that if we have parallel vectors 𝑢 and 𝑣 such that 𝑢=𝑘𝑣, where 𝑘 is a positive value, then vectors 𝑢 and 𝑣 are parallel and point in the same direction. If 𝑘 is negative, then vectors 𝑢 and 𝑣 are parallel but point in opposite directions. In the figure above, vectors 𝑢, 𝑎, and 𝑐 are parallel in the same direction, but the parallel vector 𝑏 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 𝐴=(2,1) and 𝐵=(6,3) are parallel.

  1. True
  2. False

Answer

Two vectors 𝑢 and 𝑣 are parallel if they are scalar multiples of one another, that is, 𝑢=𝑘𝑣 for any scalar 𝑘, where 𝑘0.

Therefore, we check if there is a constant 𝑘 for which 𝐴=𝑘𝐵.

Given that 𝐴=(2,1) and 𝐵=(6,3), we have (2,1)=𝑘(6,3).

Evaluating the 𝑥-components gives 2=6𝑘13=𝑘.

Taking the 𝑦-components, we have 1=3𝑘13=𝑘.

Thus, we could write that 𝐴=13𝐵, and the statement that vectors 𝑢 and 𝑣 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 𝑢=(𝑥,𝑦) and 𝑣=(𝑥,𝑦) is given by 𝑢𝑣=𝑥𝑥+𝑦𝑦.

Additionally, we can write that 𝑢𝑣=𝑢𝑣𝜃,cos where 𝜃 is the angle between 𝑢 and 𝑣.

We can use the angle definition of the dot product to help us identify perpendicular vectors. If the two vectors 𝑢 and 𝑣 are perpendicular, then the angle 𝜃=90.

Thus, we can write 𝑢𝑣=𝑢𝑣90=𝑢𝑣×0=0.cos

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 𝜃=180 or 0, then the vectors are parallel.

Definition: Perpendicular Vectors

Two vectors 𝑢=(𝑥,𝑦) and 𝑣=(𝑥,𝑦) are perpendicular if 𝑢𝑣=0.

Equivalently, by calculating the dot product, 𝑢 and 𝑣 are perpendicular if 𝑥𝑥+𝑦𝑦=0.

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?

  1. (2,0),(3,6)
  2. (1,4),(2,8)
  3. (0,7),(0,9)
  4. (3,0),(0,6)

Answer

We recall that two vectors, 𝑢=(𝑥,𝑦) and 𝑣=(𝑥,𝑦), are perpendicular if 𝑢𝑣=0.

To calculate 𝑢𝑣, we multiply the corresponding components of each vector and add the resulting numbers: 𝑢𝑣=𝑥𝑥+𝑦𝑦.

Therefore, for each pair of vectors above, we check if 𝑥𝑥+𝑦𝑦=0.

For the first vector pair, (2,0) and (3,6), we have (2,0)(3,6)=23+0(6)=6+0=6.

As 60, (2,0) and (3,6) are not perpendicular.

The dot product of the second pair of vectors in option B, (1,4) and (2,8), can be found to be (1,4)(2,8)=12+48=2+32=34.

As 340, (1,4) and (2,8) are not perpendicular.

Next, the dot product of the vectors (0,7) and (0,9) is (0,7)(0,9)=00+79=0+63=63.

Therefore, (0,7) and (0,9) are not perpendicular.

The final pair of vectors in option D, (3,0) and (0,6), have a dot product of (3,0)(0,6)=30+06=0+0=0.

As the dot product is equal to zero, (3,0) and (0,6) 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 𝐴=(,+2) and 𝐵=(3,41), then one of the values of that makes 𝐴𝐵 is .

  1. 7
  2. 5
  3. 5
  4. 7

Answer

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

We can write this as 𝐴=𝑘𝐵 for any scalar 𝑘, where 𝑘0.

We can substitute the values 𝐴=(,+2) and 𝐵=(3,41) to find the value of 𝑘, giving (,+2)=𝑘(3,41).

Evaluating the 𝑥-components gives =𝑘×3=3𝑘.

Dividing through by 3, we have 3=3𝑘313=𝑘.

We evaluate the 𝑦-components above to give +2=𝑘(41).

Substituting the value 𝑘=13 into this equation, we have +2=13(41).

Expanding the parentheses and subtracting from both sides of the equation give +2=43132=1313.

Simplifying this equation by adding 13 to both sides and then multiplying by 3, we have 73=137=.

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

We can check our answer by substituting =7 into vectors 𝐴 and 𝐵 and checking if 𝐴=𝑘𝐵. This gives (,+2)=𝑘(3,41)(7,9)=𝑘(21,27).

We can see that our vectors are scalar multiples of each other. In this case, 𝑘=13.

We check the other answer options. In option B, =5. So if 𝐴=𝑘𝐵, then we would have (,+2)=𝑘(3,41)(5,7)=𝑘(15,19).

However, there is no value of 𝑘 that would make this valid, so (5,7)𝑘(15,19) and 𝐴𝐵 when =5.

In answer option C, =5. If 𝐴𝐵, then we would have (,+2)=𝑘(3,41)(5,3)=𝑘(15,21).

However, there is no value of 𝑘 that would be valid, so (5,3)𝑘(15,21) and 𝐴𝐵 when =5.

Finally, checking answer option D by substituting =7 into (,+2)=𝑘(3,41) gives (7,5)=𝑘(21,29).

However, there is no valid value of 𝑘 here, so (7,5)𝑘(21,29) and 𝐴𝐵 when =7.

Therefore, we have shown that the only valid answer from the available options here is =7.

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 𝐴=𝑘,1, 𝐵=(2,8), and 𝐴𝐵, then 𝑘=.

  1. 2
  2. 2
  3. 2,2
  4. 4

Answer

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

This means that for any perpendicular vectors 𝐴=(𝑥,𝑦) and 𝐵=(𝑥,𝑦), 𝐴𝐵=𝑥𝑥+𝑦𝑦=0.

We are given that 𝐴=𝑘,1 and 𝐵=(2,8). Therefore, 𝐴𝐵=𝑘2+1(8)=2𝑘8.

Since 𝐴𝐵=0, we can write that 2𝑘8=0.

Simplifying by adding 8 to both sides and then dividing through by 2 gives 2𝑘=8𝑘=4.

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, 𝑘=±4=±2.

Therefore, we can give the answer that the values of 𝑘 must be those given in option C: 2,2.

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 𝐴=(1,2) and 𝐵=(2,1) are .

Answer

We may approach this question by considering if vectors 𝐴 and 𝐵 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 𝐴=𝑘𝐵 for any scalar 𝑘, where 𝑘0.

Substituting the given vectors, 𝐴=(1,2) and 𝐵=(2,1), we check if there is a value of 𝑘 for which (1,2)=𝑘(2,1).

Evaluating the 𝑥-components gives 1=2𝑘12=𝑘𝑘=12.

Evaluating the 𝑦-components gives 2=𝑘,𝑘=2.

However, as both values of 𝑘 are different, there is no single value of 𝑘 that solves the equation. So, (1,2)𝑘(2,1).

Therefore, vectors 𝐴 and 𝐵 are not parallel.

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

The dot product of two vectors 𝑢=(𝑥,𝑦) and 𝑣=(𝑥,𝑦) is given by 𝑢𝑣=𝑥𝑥+𝑦𝑦.

Therefore, the dot product of 𝐴=(1,2) and 𝐵=(2,1) is given by 𝐴𝐵=1(2)+21=2+2=0.

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

Therefore, we can answer the question: vectors 𝐴=(1,2) and 𝐵=(2,1) are perpendicular.

We now summarize the key points.

Key Points

  • Vectors 𝑢 and 𝑣 are parallel if 𝑢=𝑘𝑣 for any scalar 𝑘, where 𝑘0.
  • Vectors 𝑢=(𝑥,𝑦) and 𝑣=(𝑥,𝑦) are perpendicular if their dot product is 0: 𝑢𝑣=𝑥𝑥+𝑦𝑦=0.

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