Lesson Explainer: Parallel and Perpendicular Vectors in Space Mathematics

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

A vector in space is defined by two quantities: its magnitude and its direction. A special relationship forms between two or more vectors when they point in the same direction or in opposite directions. When this is the case, we say that the vectors are parallel. This can be represented mathematically.

Definition: Parallel Vectors in Space

The vectors ⃑𝐴 and ⃑𝐡 are parallel if, and only if, they are scalar multiples of one another: ⃑𝐴=π‘˜βƒ‘π΅, where π‘˜ is a nonzero real number.

Another way of thinking about this is that if two vectors are parallel, then the ratios of each of their corresponding components are the same. So, if we have our two vectors ⃑𝐴=ο€Ήπ‘Ž,π‘Ž,π‘Žο…ο—ο˜ο™ and ⃑𝐡=𝑏,𝑏,π‘ο…ο—ο˜ο™, then π‘Žπ‘=π‘Žπ‘=π‘Žπ‘.ο—ο—ο˜ο˜ο™ο™

The second special relationship that can occur between two vectors is when the directions of the two vectors form a 90∘ angle. When this happens, we say that the two vectors are perpendicular to one another. In order to identify when two vectors are perpendicular, we can use the dot product.

Definition: The Dot Product

The dot products of two vectors, ⃑𝐴 and ⃑𝐡, can be defined as ⃑𝐴⋅⃑𝐡=β€–β€–βƒ‘π΄β€–β€–β€–β€–βƒ‘π΅β€–β€–πœƒ,cos where πœƒ is the angle formed between ⃑𝐴 and ⃑𝐡.

In the case that the vectors ⃑𝐴 and ⃑𝐡 are perpendicular, πœƒ=90∘. Therefore, cosπœƒ=0, and so ⃑𝐴⋅⃑𝐡=0.

Note:

When two vectors are parallel, the angle between them is either 0∘ or 180∘.

Another way in which we can define the dot product of two vectors ⃑𝐴=ο€Ήπ‘Ž,π‘Ž,π‘Žο…ο—ο˜ο™ and ⃑𝐡=𝑏,𝑏,π‘ο…ο—ο˜ο™ is by the formula ⃑𝐴⋅⃑𝐡=π‘Žπ‘+π‘Žπ‘+π‘Žπ‘.ο—ο—ο˜ο˜ο™ο™

Of course, if the vectors are perpendicular, then this sum of the products of the corresponding components will equal zero: π‘Žπ‘+π‘Žπ‘+π‘Žπ‘=0.ο—ο—ο˜ο˜ο™ο™

Definition: Perpendicular Vectors in Space

The vectors ⃑𝐴 and ⃑𝐡 are perpendicular if, and only if, their dot product is equal to zero: ⃑𝐴⋅⃑𝐡=0.

Let us now look at some examples involving parallel and perpendicular vectors.

Example 1: Using the Properties of Parallel and Perpendicular Vectors to Solve a Problem

True or False: If the component of a vector in the direction of another vector is zero, then the two are parallel.

Answer

In order to visualize what is going on here, let us start by considering two vectors, ⃑𝐴 and ⃑𝐡. These can be any two arbitrary vectors. Let us suppose that these vectors start at the same point. Here is what they may look like.

Now, let us add in the component of ⃑𝐴 in the direction of ⃑𝐡.

Using this diagram, along with some right trigonometry, we can see that the magnitude of the component of ⃑𝐴 in the direction of ⃑𝐡 can be calculated using β€–β€–βƒ‘π΄β€–β€–πœƒcos.

The question states that this quantity must be equal to zero. Since ‖‖⃑𝐴‖‖≠0, cosπœƒ must be equal to zero. Solving this for πœƒ gives us πœƒ=90.∘

This tells us that when the component of ⃑𝐴 in the direction of ⃑𝐡 is 0, the angle between the vectors must be a right angle. Therefore, ⃑𝐴 and ⃑𝐡 are perpendicular to one another.

Our solution to the question is β€œFalse”: if the component of a vector in the direction of another vector is zero, then the two are not parallel.

Another way that we can think about this problem is by picturing two parallel vectors as shown.

Since the vectors ⃑𝐴 and ⃑𝐡 are parallel, the angle between the two vectors is πœƒ=0∘. Therefore, the component of ⃑𝐴 in the direction of ⃑𝐡 is ‖‖⃑𝐴‖‖(0)cos∘. Since cos(0)=1∘, this component is just the vector ⃑𝐴. ⃑𝐴 is nonzero; therefore, the answer to the question must be β€œFalse”.

In the next example, we will find missing parameters such that the two given vectors are parallel.

Example 2: Finding a Missing Value Using a Pair of Parallel Vector

Find the values of π‘š and 𝑛 so that vector 2⃑𝑖+7⃑𝑗+π‘šβƒ‘π‘˜ is parallel to vector 6⃑𝑖+π‘›βƒ‘π‘—βˆ’21βƒ‘π‘˜.

Answer

In order to solve this problem, we can use the fact that when two vectors are parallel to one another, they are scalar multiples of one another. Therefore, 2⃑𝑖+7⃑𝑗+π‘šβƒ‘π‘˜=𝑐6⃑𝑖+𝑛⃑𝑗+21βƒ‘π‘˜ο‡, where 𝑐 is just a constant that can be found.

By equating the coefficients of each of the vector components, we end up with three equations: 2=6𝑐,7=𝑐𝑛,π‘š=βˆ’21𝑐.

We can solve the first of these equations to find 𝑐. In doing this, we obtain 𝑐=13.

Now, we simply need to substitute this value into the other two equations and solve for the missing values. To find 𝑛, we have 7=𝑛3𝑛=21.

To find π‘š, we have π‘š=βˆ’213π‘š=βˆ’7.

We have now reached our solution, which is that the values of π‘š and 𝑛 that make the vectors parallel are π‘š=βˆ’7 and 𝑛=21.

We can check our solution by checking that the ratios of corresponding components of the two vectors are equal.

For two parallel vectors, it should be true that 26=7𝑛=π‘šβˆ’21.

If we substitute in the values we obtained, we get 26=721=βˆ’7βˆ’21, which all simplify down to 13. This confirms that our solution is correct.

In the next example, we will look at how we can identify perpendicular vectors.

Example 3: Determining the Vector That is Not Perpendicular to the Given Line

Which of the following vectors is not perpendicular to the line whose direction vector βƒ‘π‘Ÿ is (2,βˆ’3,5)?

  1. (10,10,2)
  2. (βˆ’10,βˆ’5,1)
  3. (2,βˆ’2,βˆ’2)
  4. (1,βˆ’2,3)
  5. (2,3,1)

Answer

In order for two vectors to be perpendicular to one another, it must be true that their dot product is equal to zero. In order to find the solution, we just need to find which of the vectors does not give zero when dotted with (2,βˆ’3,5).

Let us start with the vector in A. We find that the dot product gives us (10,10,2)β‹…(2,βˆ’3,5)=10Γ—2+10Γ—(βˆ’3)+2Γ—5=20βˆ’30+10=0.

Therefore, (10,10,2) is perpendicular to (2,βˆ’3,5).

Next, we can check the vector in B. We obtain (βˆ’10,βˆ’5,1)β‹…(2,βˆ’3,5)=(βˆ’10)Γ—2+(βˆ’5)Γ—(βˆ’3)+1Γ—5=βˆ’20+15+5=0.

Since this is zero, the answer is also not B.

Dotting the vector in C with (2,βˆ’3,5), we obtain (2,βˆ’2,βˆ’2)β‹…(2,βˆ’3,5)=2Γ—2+(βˆ’2)Γ—(βˆ’3)+(βˆ’2)Γ—5=4+6βˆ’10=0.

So, the solution cannot be C either.

Next, we need to check the vector D: (1,βˆ’2,3)β‹…(2,βˆ’3,5)=1Γ—2+(βˆ’2)Γ—(βˆ’3)+3Γ—5=2+6+15=23.

Since this dot product is nonzero, (1,βˆ’2,3) and (2,βˆ’3,5) are not perpendicular. So, our solution to the question is that the vector that is not perpendicular to the line is D, (1,βˆ’2,3).

We can quickly check the vector E, just to make sure that this vector is perpendicular to the line: (2,3,1)β‹…(2,βˆ’3,5)=2Γ—2+3Γ—(βˆ’3)+1Γ—5=4βˆ’9+5=0.

Since this result is zero, this helps to confirm our solution: D.

In the next example, we will be determining whether two vectors are parallel, perpendicular, or neither.

Example 4: Identifying Whether Two Vectors are Parallel, Perpendicular, or Otherwise

Given the two vectors ⃑𝐴=(8βƒ‘π‘–βˆ’7⃑𝑗+βƒ‘π‘˜) and ⃑𝐡=(64βƒ‘π‘–βˆ’56⃑𝑗+8βƒ‘π‘˜), determine whether these two vectors are parallel, perpendicular, or otherwise.

Answer

Let us start by recalling the conditions under which these vectors are either parallel or perpendicular. The vectors are parallel if ⃑𝐴=π‘˜βƒ‘π΅, where π‘˜ is a nonzero real constant. The vectors are perpendicular if ⃑𝐴⋅⃑𝐡=0. If neither of these conditions are met, then the vectors are neither parallel nor perpendicular to one another.

Let us start by checking whether they are parallel. If they are parallel, then it would be true that (8βƒ‘π‘–βˆ’7⃑𝑗+βƒ‘π‘˜)=π‘˜(64βƒ‘π‘–βˆ’56⃑𝑗+8βƒ‘π‘˜).

We can check whether this is true by trying to find a value of π‘˜. We can form three equations by equating the components of these vectors. We obtain 8=64π‘˜,βˆ’7=βˆ’56π‘˜,1=8π‘˜.

By solving any of these equations, we will get the same value of π‘˜=18. This tells us that our vectors are in fact parallel.

We have already reached our solution, but, to illustrate the method, let us complete the proof that they are not perpendicular. We find the dot product of ⃑𝐴 and ⃑𝐡 to be (8βƒ‘π‘–βˆ’7⃑𝑗+βƒ‘π‘˜)β‹…(64βƒ‘π‘–βˆ’56⃑𝑗+8βƒ‘π‘˜)=8Γ—64+(βˆ’7)Γ—(βˆ’56)+1Γ—8=512+392+8=912.

Since this is not equal to zero, the vectors are not perpendicular, which does not contradict our solution that the vectors are parallel.

Now we know how to work out whether two vectors are parallel or perpendicular. We can use this to work out whether two straight lines are parallel or perpendicular. In the next example, we will see how we can find a missing constant in the equation of a straight line given another line that is perpendicular to it.

Example 5: Solving a Problem Involving a Pair of Perpendicular Straight Lines

If the straight line π‘₯+8βˆ’10=𝑦+8π‘š=𝑧+10βˆ’8 is perpendicular to π‘₯+5βˆ’4=𝑦+810 and 𝑧=8, find π‘š.

Answer

So, we have been given the equations of two straight lines in 3D space and told that they are perpendicular to one another. We can use this information to find the missing constant.

First, we need to find the direction vector of each line. We know that a line of the form π‘₯βˆ’π‘₯π‘Ž=π‘¦βˆ’π‘¦π‘=π‘§βˆ’π‘§π‘οŠ§οŠ§οŠ§ passes through the point (π‘₯,𝑦,𝑧) and has a direction vector of (π‘Ž,𝑏,𝑐).

Using this, we can see that the line π‘₯+8βˆ’10=𝑦+8π‘š=𝑧+10βˆ’8 has a direction vector of (βˆ’10,π‘š,βˆ’8).

The other line is not quite of this form, because we have 𝑧=8. However, this just means that the 𝑧-coordinate of this line is constant since our line exists in the plane 𝑧=8. So, in the direction vector, the 𝑧-component will be zero. Therefore, the direction vector of the line π‘₯+5βˆ’14=𝑦+810, with 𝑧=8, is (βˆ’4,10,0).

Now that we have the direction vectors of the two lines, we need to take the dot product of these vectors and set it to be equal to zero since the two lines are perpendicular to one another. This gives us (βˆ’10,π‘š,βˆ’8)β‹…(βˆ’4,10,0)=0.

All we need to do is solve this to find π‘š. We obtain (βˆ’10)Γ—(βˆ’4)+π‘šΓ—10+(βˆ’8)Γ—0=040+10π‘š=010π‘š=βˆ’40π‘š=βˆ’4.

Here, we have reached our solution: in order for these two lines to be perpendicular, π‘š=βˆ’4.

We have now seen a variety of examples of how we can find and use parallel and perpendicular vectors. Let us recap some key points of the explainer.

Key Points

  • When two vectors are parallel, the angle between them is 0∘ or 180∘. When two vectors are perpendicular, the angle between them is 90∘.
  • Two vectors, ⃑𝐴=ο€Ήπ‘Ž,π‘Ž,π‘Žο…ο—ο˜ο™ and ⃑𝐡=𝑏,𝑏,π‘ο…ο—ο˜ο™, are parallel if ⃑𝐴=π‘˜βƒ‘π΅. This is equivalent to the ratios of the corresponding components of each of the vectors being equal: π‘Žπ‘=π‘Žπ‘=π‘Žπ‘.ο—ο—ο˜ο˜ο™ο™
  • Two vectors, ⃑𝐴=ο€Ήπ‘Ž,π‘Ž,π‘Žο…ο—ο˜ο™ and ⃑𝐡=𝑏,𝑏,π‘ο…ο—ο˜ο™, are perpendicular if their dot product is zero, ⃑𝐴⋅⃑𝐡=0, or if π‘Žπ‘+π‘Žπ‘+π‘Žπ‘=0.ο—ο—ο˜ο˜ο™ο™
  • We can use the above relationships between parallel and perpendicular vectors to determine whether two lines are parallel or perpendicular.

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