Lesson Explainer: Angle between Two Vectors in Space | Nagwa Lesson Explainer: Angle between Two Vectors in Space | Nagwa

Lesson Explainer: Angle between Two Vectors in Space Mathematics

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, 𝑢=(𝑢,𝑢,𝑢,,𝑢) and 𝑣=(𝑣,𝑣,𝑣,,𝑣), then the dot product of the two vectors is equal to 𝑢𝑣=𝑢𝑣+𝑢𝑣+𝑢𝑣++𝑢𝑣, which is a scalar quantity. Also, we recall the following properties of the dot product.

Theorem: Properties of the Dot Product

For any scalar 𝑐 and vectors 𝑢, 𝑣, and 𝑤 of the same dimension, the following properties hold: 𝑢𝑣=𝑣𝑢(),𝑐𝑢𝑣=𝑐𝑢𝑣(),𝑢𝑣+𝑤=𝑢𝑣+𝑢𝑤(),𝑢𝑢=𝑢.thecommutativepropertythescalarmultiplicationpropertythedistributiveproperty

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 𝑢+𝑣2𝑢𝑣𝜃=𝑢𝑣.cos

Using the last property from the list, we can rewrite the right-hand side of the equation as 𝑢𝑣𝑢𝑣. Using the distributive property of the dot product, we get 𝑢𝑣𝑢𝑣=𝑢𝑢𝑣𝑣𝑢𝑣=𝑢𝑢𝑢𝑣𝑣𝑢𝑣𝑣.

We know that 𝑢𝑢=𝑢 and 𝑣𝑣=𝑣. By the commutative property, we have 𝑢𝑣=𝑣𝑢. Applying these identities to the right-hand side of the equation above reduces it to 𝑢+𝑣2𝑢𝑣.

This leads to the equation 𝑢+𝑣2𝑢𝑣𝜃=𝑢+𝑣2𝑢𝑣.cos

Simplifying this equation, we get 2𝑢𝑣𝜃=2𝑢𝑣,𝑢𝑣𝜃=𝑢𝑣.coscos

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 𝑢 and 𝑣 be nonzero vectors, and let 𝜃 be the angle between the two vectors. Then, 𝑢𝑣=𝑢𝑣𝜃.cos

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 𝐴 and 𝐵 is 22. If 𝐴=3𝐵=25.2, find 𝐴𝐵 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, 𝐴𝐵=𝐴𝐵𝜃,cos where 𝜃 is the angle between the two vectors. We are given that 𝜃=22 and 𝐴=25.2. We can compute 𝐵=25.23=8.4.

Then, 𝐴𝐵=25.2×8.4×22196.266.cos

Rounding to the nearest hundredth, we get 𝐴𝐵=196.27.

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 𝐴 and 𝐵 are two perpendicular unit vectors, find 3𝐴𝐵2𝐴+𝐵.

Answer

We recall the distributive property of the dot product; for any vectors of equal dimension, 𝑢, 𝑣, and 𝑤: 𝑢𝑣+𝑤=𝑢𝑣+𝑢𝑤.

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

3𝐴𝐵2𝐴+𝐵=3𝐴𝐵2𝐴+3𝐴𝐵𝐵.(1)

We also recall that the commutative property of the dot product; for any vectors of equal dimension, 𝑢 and 𝑣: 𝑢𝑣=𝑣𝑢.

Then, the right-hand side of equation (1) can be written as 2𝐴3𝐴𝐵+𝐵3𝐴𝐵.

Using the distributive property again, this is equal to

2𝐴3𝐴2𝐴𝐵+𝐵3𝐴𝐵𝐵.(2)

Next, we recall the scalar multiplication property of the dot product; for any scalar 𝑐 and vectors of equal dimension, 𝑢 and 𝑣: 𝑐𝑢𝑣=𝑢𝑐𝑣=𝑐𝑢𝑣.

Then expression (2) is equal to

6𝐴𝐴+2𝐴𝐵+3𝐵𝐴𝐵𝐵.(3)

Finally, we recall that, for any vector 𝑢, 𝑢𝑢=𝑢.

Using this property and the commutative property of the dot product, expression (3) can be written as 6𝐴+2𝐴𝐵+3𝐴𝐵𝐵,6𝐴+5𝐴𝐵𝐵.whichsimpliesto

We know that the magnitudes 𝐴 and 𝐵 are both equal to 1, since they are given to be unit vectors. So, what remains is to compute the dot product 𝐴𝐵. We recall that 𝐴𝐵=𝐴𝐵𝜃,cos where 𝜃 is the angle between the two vectors. Since we are given that the two vectors are perpendicular, we have 𝜃=90. Then, 𝐴𝐵=1×1×90=0.cos

We can then substitute these values into our expression: 6𝐴+5𝐴𝐵𝐵=6×1+5×01=7.

Hence, 3𝐴𝐵2𝐴+𝐵=7.

We note that there are two different ways to measure the angle 𝜃 between any two vectors, 𝑢 and 𝑣, 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 𝑢 and 𝑣 having the same initial point.

The two angles 𝜃 and 𝜃 satisfy the equation 𝜃+𝜃=360, which means 𝜃=360𝜃. The geometric formula in the theorem holds for both 𝜃 and 𝜃, because cosine is even and periodic with a period of 360. More specifically, we have coscoscoscos𝜃=(𝜃)=(𝜃+360)=𝜃.

We note that one of the angles (𝜃 in the picture above) lies between 0 and 180, while the other angle (𝜃 from the picture) lies between 180 and 360. 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 0 and 180.

We discussed a geometric formula for the dot product: 𝑢𝑣=𝑢𝑣𝜃.cos

To calculate the angle between two vectors, we can rearrange this equation so that 𝜃 is the subject of the equation. If 𝑢 and 𝑣 are nonzero vectors, then 𝑢0 and 𝑣0 so we can divide both sides of the equation by 𝑢𝑣. Then, coscos𝜃=𝑢𝑣𝑢𝑣𝜃=𝑢𝑣𝑢𝑣.

We recall that the inverse cosine function has a range between 0 and 180, which is also where the angle between two vectors is defined.

Theorem: Angle between Two Vectors

Let 𝑢 and 𝑣 be nonzero vectors. Then, the angle 𝜃 between the two vectors, which lies between 0 and 180, is given by 𝜃=𝑢𝑣𝑢𝑣.cos

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 𝐴=35, 𝐵=23, and 𝐴𝐵=80522, determine the measure of the smaller angle between the two vectors.

Answer

We recall that the angle 𝜃 between any two nonzero vectors 𝐴 and 𝐵 is given by 𝜃=𝐴𝐵𝐴𝐵.cos

We are given that 𝐴=35, 𝐵=23, and 𝐴𝐵=80522, so we have 𝜃=35×23=80521610=22=135.coscoscos

Hence, the angle between 𝐴 and 𝐵 is 135. We observe that the answer is between 0 and 180, 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 𝐴=2𝑖+5𝑘 and 𝐵=4𝑖+3𝑗+𝑘, 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 𝐴 and 𝐵 is the angle 𝜃 between 0 and 180 satisfying cos𝜃=𝐴𝐵𝐴𝐵.

Given a vector 𝑣=𝑣𝑖+𝑣𝑗+𝑣𝑘, we know that 𝑣=𝑣+𝑣+𝑣.

Since 𝐴=2𝑖+5𝑘 and 𝐵=4𝑖+3𝑗+𝑘, we can compute 𝐴=2+0+5=29,𝐵=4+3+1=26.

Also, given any two vectors 𝑢=𝑢𝑖+𝑢𝑗+𝑢𝑘 and 𝑣=𝑣𝑖+𝑣𝑗+𝑣𝑘, the dot product of the two vectors is equal to 𝑢𝑣=𝑢𝑣+𝑢𝑣+𝑢𝑣.

So, 𝐴𝐵=2×4+0×3+5×1=13.

As noted, angle 𝜃 must satisfy cos𝜃=1329×26=0.4734.

This leads to 𝜃=(0.4734)=61.742.cos

Hence, the measure of the angle between the two given vectors rounded to the nearest hundredth is 61.74. We observe that the answer is between 0 and 180, 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 𝜃 between the vectors 𝑉=𝑖+2𝑗+𝑘 and 𝑊=3𝑖+6𝑗+3𝑘.

Answer

For this example, we can use two different methods to find the angle between 𝑉 and 𝑊. 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 𝑉 and 𝑊 is given by 𝜃=𝑉𝑊𝑉𝑊.cos

Since 𝑉=𝑖+2𝑗+𝑘 and 𝑊=3𝑖+6𝑗+3𝑘, we can calculate 𝑉=(1)+2+1=6,𝑊=(3)+6+3=54.

Their dot product is given by 𝑉𝑊=(1)×(3)+2×6+1×3=18.

Then, angle 𝜃 is given by 𝜃=𝑉𝑊𝑉𝑊=18654=(1)=0.coscoscos

So, the angle between 𝑉 and 𝑊 is 0.

Method 2

We recall that two nonzero vectors 𝑉 and 𝑊 are parallel if there is a scalar 𝑐0 satisfying 𝑉=𝑐𝑊.

Furthermore, if 𝑐>0, then the two vectors have the same direction. In this case, the angle between the two vectors is 0. On the other hands, if 𝑐<0, then the two vectors are pointing in opposite directions, meaning that the angle between them is 180.

We are given that 𝑉=𝑖+2𝑗+𝑘 and 𝑊=3𝑖+6𝑗+3𝑘. We note that each coordinate of 𝑊 is precisely 3 times the corresponding coordinate of 𝑉. In other words, 𝑊=3𝑖+6𝑗+3𝑘=3𝑖+2𝑗+𝑘=3𝑉.

So, 𝑊=3𝑉, meaning that the vectors 𝑉 and 𝑊 are parallel. Since the scalar 3 is positive, this means that they have the same direction.

Hence, the angle between the two vectors is 0.

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 𝑣 and 𝑤 is the angle 𝜃=𝑣𝑤𝑣𝑤.cos

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 (1,0,3). The purple vector has its terminal point at (1,2,0), while the red vector has the terminal point at (0,2,0). Then, the purple vector is given by (1,2,0)(1,0,3)=(0,2,3).

The red vector is given by (0,2,0)(1,0,3)=(1,2,3).

We compute their magnitudes and the dot product for the formula of the angle between two vectors: (0,2,3)=0+2+(3)=13,(1,2,3)=(1)+2+(3)=14,(0,2,3)(1,2,3)=0×(1)+2×2+(3)×(3)=13.

Hence, the angle between the two vectors is given by 𝜃=1313×14=(0.9636)=15.51.coscos

So, the angle between the two given vectors rounded to the nearest degree is 16. We observe that both values lie between 0 and 180, 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 𝐴(3,5,6), 𝐵(0,3,7), 𝐶(8,10,2), and 𝐷(3,9,6), determine the measure of the angle between vectors 𝐴𝐵 and 𝐶𝐷 rounded to the nearest hundredth.

Answer

We recall that the angle between any two nonzero vectors 𝑣 and 𝑤 is the angle 𝜃=𝑣𝑤𝑣𝑤.cos

We need to identify the vectors 𝐴𝐵 and 𝐶𝐷 before computing their magnitudes and the dot product. We have 𝐴𝐵=(0(3),3(5),76)=(3,8,13),𝐶𝐷=(3(8),910,6(2))=(5,1,4).

Then, we compute 𝐴𝐵=3+8+(13)=242,𝐶𝐷=5+(1)+(4)=42,𝐴𝐵𝐶𝐷=3×5+8×(1)+(13)×(4)=59.

Hence, we get 𝜃=59242×42=54.181.cos which, to the nearest hundredth, is 54.18.

The measure of the angle between vectors 𝐴𝐵 and 𝐶𝐷 rounded to the nearest hundredth is 54.18. We observe that the answer is between 0 and 180, which is the correct range.

Let us summarize a few important concepts from this explainer.

Key Points

  • The dot product of two nonzero vectors 𝑢 and 𝑣 is given by 𝑢𝑣=𝑢𝑣𝜃,cos where 𝜃 is the angle between the two vectors.
  • The angle 𝜃 between two nonzero vectors 𝑢 and 𝑣 is given by 𝜃=𝑢𝑣𝑢𝑣.cos
  • By convention, the angle between two vectors refers to the smallest nonnegative angle between these two vectors, which is the one between 0 and 180.
  • If the angle between two vectors is either 0 or 180, then the vectors are parallel.

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