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 and are parallel if, and only if, they are scalar multiples of one another: where 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 and , then
The second special relationship that can occur between two vectors is when the directions of the two vectors form a 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, and , can be defined as where is the angle formed between and .
In the case that the vectors and are perpendicular, . Therefore, , and so .
When two vectors are parallel, the angle between them is either or .
Another way in which we can define the dot product of two vectors and is by the formula
Of course, if the vectors are perpendicular, then this sum of the products of the corresponding components will equal zero:
Definition: Perpendicular Vectors in Space
The vectors and are perpendicular if, and only if, their dot product is equal to zero:
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.
In order to visualize what is going on here, let us start by considering two vectors, and . 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 in the direction of .
Using this diagram, along with some right trigonometry, we can see that the magnitude of the component of in the direction of can be calculated using .
The question states that this quantity must be equal to zero. Since , must be equal to zero. Solving this for gives us
This tells us that when the component of in the direction of is 0, the angle between the vectors must be a right angle. Therefore, and are perpendicular to one another.
Our solution to the question is “False”: 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 and are parallel, the angle between the two vectors is . Therefore, the component of in the direction of is . Since , this component is just the vector . is nonzero; therefore, the answer to the question must be “False”.
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 and so that vector is parallel to vector .
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, where is just a constant that can be found.
By equating the coefficients of each of the vector components, we end up with three equations:
We can solve the first of these equations to find . In doing this, we obtain
Now, we simply need to substitute this value into the other two equations and solve for the missing values. To find , we have
To find , we have
We have now reached our solution, which is that the values of and that make the vectors parallel are and .
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
If we substitute in the values we obtained, we get which all simplify down to . 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 is ?
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 .
Let us start with the vector in A. We find that the dot product gives us
Therefore, is perpendicular to .
Next, we can check the vector in B. We obtain
Since this is zero, the answer is also not B.
Dotting the vector in C with , we obtain
So, the solution cannot be C either.
Next, we need to check the vector D:
Since this dot product is nonzero, and are not perpendicular. So, our solution to the question is that the vector that is not perpendicular to the line is D, .
We can quickly check the vector E, just to make sure that this vector is perpendicular to the line:
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 and , determine whether these two vectors are parallel, perpendicular, or otherwise.
Let us start by recalling the conditions under which these vectors are either parallel or perpendicular. The vectors are parallel if , where is a nonzero real constant. The vectors are perpendicular if . 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
We can check whether this is true by trying to find a value of . We can form three equations by equating the components of these vectors. We obtain
By solving any of these equations, we will get the same value of . 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 and to be
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 is perpendicular to and , find .
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 passes through the point and has a direction vector of .
Using this, we can see that the line has a direction vector of
The other line is not quite of this form, because we have . However, this just means that the -coordinate of this line is constant since our line exists in the plane . So, in the direction vector, the -component will be zero. Therefore, the direction vector of the line , with , is
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
All we need to do is solve this to find . We obtain
Here, we have reached our solution: in order for these two lines to be perpendicular, .
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.
- When two vectors are parallel, the angle between them is or . When two vectors are perpendicular, the angle between them is .
- Two vectors, and , are parallel if This is equivalent to the ratios of the corresponding components of each of the vectors being equal:
- Two vectors, and , are perpendicular if their dot product is zero, or if
- We can use the above relationships between parallel and perpendicular vectors to determine whether two lines are parallel or perpendicular.