In this explainer, we will learn how to find the angle between two vectors in space using their dot product.
To start, let us recall how to calculate the dot product (or scalar product) of two vectors in space. If we consider two vectors with equal dimension, and , then the dot product of the two vectors is equal to which is a scalar quantity. Also, we recall the following properties of the dot product.
Theorem: Properties of the Dot Product
For any scalar and vectors , , and of the same dimension, the following properties hold:
Let us examine the geometric meaning of the dot product by using the picture below.
Using the notations from the figure, the law of cosines states that
Using the last property from the list, we can rewrite the right-hand side of the equation as . Using the distributive property of the dot product, we get
We know that and . By the commutative property, we have . Applying these identities to the right-hand side of the equation above reduces it to
This leads to the equation
Simplifying this equation, we get
Hence, the dot product of two vectors is, geometrically, the product of the magnitudes of the two vectors multiplied by the cosine of the angle between them.
Theorem: Geometric Formula for the Dot Product
Let and be nonzero vectors, and let be the angle between the two vectors. Then,
Let us consider an example where we apply this geometric formula to compute the dot product.
Example 1: Finding the Dot Product between Vectors
The angle between and is . If , find to the nearest hundredth.
Answer
We recall that the dot product of two vectors is the product of the magnitudes of the two vectors multiplied by the cosine of the angle between them. In other words, where is the angle between the two vectors. We are given that and . We can compute
Then,
Rounding to the nearest hundredth, we get .
In the next example, we use the properties of vector operations together with the geometric interpretation of the dot product.
Example 2: Finding the Dot Product between Vectors Using the Properties of Dot Product
If and are two perpendicular unit vectors, find .
Answer
We recall the distributive property of the dot product; for any vectors of equal dimension, , , and :
We remark that the distributive property works the same when the plus signs on both sides are replaced by minus signs. Using this property, we compute
We also recall that the commutative property of the dot product; for any vectors of equal dimension, and :
Then, the right-hand side of equation (1) can be written as
Using the distributive property again, this is equal to
Next, we recall the scalar multiplication property of the dot product; for any scalar and vectors of equal dimension, and :
Then expression (2) is equal to
Finally, we recall that, for any vector ,
Using this property and the commutative property of the dot product, expression (3) can be written as
We know that the magnitudes and are both equal to 1, since they are given to be unit vectors. So, what remains is to compute the dot product . We recall that where is the angle between the two vectors. Since we are given that the two vectors are perpendicular, we have . Then,
We can then substitute these values into our expression:
Hence, .
We note that there are two different ways to measure the angle between any two vectors, and , as pictured below. Remember, a vector is a quantity with magnitude and direction, which we can sketch starting anywhere in space. This means we can sketch both and having the same initial point.
The two angles and satisfy the equation , which means . The geometric formula in the theorem holds for both and , because cosine is even and periodic with a period of . More specifically, we have
We note that one of the angles ( in the picture above) lies between and , while the other angle ( from the picture) lies between and . By convention, when we say the angle between two vectors we mean the smallest nonnegative angle between these two vectors, which is the one between and .
We discussed a geometric formula for the dot product:
To calculate the angle between two vectors, we can rearrange this equation so that is the subject of the equation. If and are nonzero vectors, then and so we can divide both sides of the equation by . Then,
We recall that the inverse cosine function has a range between and , which is also where the angle between two vectors is defined.
Theorem: Angle between Two Vectors
Let and be nonzero vectors. Then, the angle between the two vectors, which lies between and , is given by
In the next example, we compute the angle between two vectors, given their magnitudes and the dot product.
Example 3: Finding the Measure of the Smaller Angle between Two Vectors Given Their Magnitudes and Their Dot Product
Given that , , and , determine the measure of the smaller angle between the two vectors.
Answer
We recall that the angle between any two nonzero vectors and is given by
We are given that , , and , so we have
Hence, the angle between and is . We observe that the answer is between and , which is the correct range.
In the next example, we will compute the angle between two vectors given in terms of the unit directional vectors.
Example 4: Finding the Angle between Vectors given in terms of Fundamental Unit Vectors
If and , find the measure of the angle between the two vectors rounded to the nearest hundredth.
Answer
We recall that the angle between any two nonzero vectors and is the angle between and satisfying
Given a vector , we know that
Since and , we can compute
Also, given any two vectors and , the dot product of the two vectors is equal to
So,
As noted, angle must satisfy
This leads to
Hence, the measure of the angle between the two given vectors rounded to the nearest hundredth is . We observe that the answer is between and , which is the correct range.
In the next example, we compute the angle between two parallel vectors.
Example 5: Finding the Angle between Two Given Vectors in a Three-Dimensional Plane
Find the angle between the vectors and .
Answer
For this example, we can use two different methods to find the angle between and . The first method is to use the dot product to find the angle between two vectors, and the second method is to use the property of parallel vectors.
Method 1
We recall that the angle between any two nonzero vectors and is given by
Since and , we can calculate
Their dot product is given by
Then, angle is given by
So, the angle between and is .
Method 2
We recall that two nonzero vectors and are parallel if there is a scalar satisfying
Furthermore, if , then the two vectors have the same direction. In this case, the angle between the two vectors is . On the other hands, if , then the two vectors are pointing in opposite directions, meaning that the angle between them is .
We are given that and . We note that each coordinate of is precisely 3 times the corresponding coordinate of . In other words,
So, , meaning that the vectors and are parallel. Since the scalar 3 is positive, this means that they have the same direction.
Hence, the angle between the two vectors is .
In our next example, we identify the angle between two vectors that are given graphically.
Example 6: Finding the Angle between Two Given Vectors from Graphs
Find the measure of the angle between the two vectors shown in the figure. Round your answer to the nearest degree.
Answer
We recall that the angle between any two nonzero vectors and is the angle
In the provided figure, we are given two vectors graphically. We will start by finding the components of the vectors from the diagrams.
We note that both vectors begin at the point . The purple vector has its terminal point at , while the red vector has the terminal point at . Then, the purple vector is given by
The red vector is given by
We compute their magnitudes and the dot product for the formula of the angle between two vectors:
Hence, the angle between the two vectors is given by
So, the angle between the two given vectors rounded to the nearest degree is . We observe that both values lie between and , which is the correct range.
In our last example, we will compute the angle between two vectors, given their endpoints.
Example 7: Determining the Angle between Vectors
Given , , , and , determine the measure of the angle between vectors and rounded to the nearest hundredth.
Answer
We recall that the angle between any two nonzero vectors and is the angle
We need to identify the vectors and before computing their magnitudes and the dot product. We have
Then, we compute
Hence, we get which, to the nearest hundredth, is .
The measure of the angle between vectors and rounded to the nearest hundredth is . We observe that the answer is between and , which is the correct range.
Let us summarize a few important concepts from this explainer.
Key Points
- The dot product of two nonzero vectors and is given by where is the angle between the two vectors.
- The angle between two nonzero vectors and is given by
- By convention, the angle between two vectors refers to the smallest nonnegative angle between these two vectors, which is the one between and .
- If the angle between two vectors is either or , then the vectors are parallel.
