Learning Outcomes
- Find the dot product of two vectors.
- Find the angle between two vectors.
Finding the Dot Product of Two Vectors
As we discussed in the previous section, scalar multiplication involves multiplying a vector by a scalar, and the result is a vector. As we have seen, multiplying a vector by a number is called scalar multiplication. If we multiply a vector by a vector, there are two possibilities: the dot product and the cross product. We will only examine the dot product here; you may encounter the cross product in more advanced mathematics courses.
The dot product of two vectors involves multiplying two vectors together, and the result is a scalar.
A General Note: Dot Product
The dot product of two vectors [latex]\boldsymbol{v}=\langle a,b\rangle [/latex] and [latex]\boldsymbol{v}=\langle c,d\rangle [/latex] is the sum of the product of the horizontal components and the product of the vertical components.
[latex]\boldsymbol{v}\cdot \boldsymbol{u}=ac+bd[/latex]
To find the angle between the two vectors, use the formula below.
[latex]\cos \theta =\dfrac{\boldsymbol{v}}{|\boldsymbol{v}|}\cdot \dfrac{\boldsymbol{u}}{|\boldsymbol{u}|}[/latex]
Example 1: Finding the Dot Product of Two Vectors
Find the dot product of [latex]\boldsymbol{v}=\langle 5,12\rangle [/latex] and [latex]\boldsymbol{u}=\langle -3,4\rangle [/latex].
Example 2: Finding the Dot Product of Two Vectors and the Angle between Them
Find the dot product of v1 = 5i + 2j and v2 = 3i + 7j. Then, find the angle between the two vectors.
Example 3: Finding the Angle between Two Vectors
Find the angle between [latex]\boldsymbol{u}=\langle -3,4\rangle [/latex] and [latex]\boldsymbol{v}=\langle 5,12\rangle [/latex].
Try It
Find the dot product of [latex]\boldsymbol{u}=[/latex] 4i – 3j and [latex]\boldsymbol{v}=[/latex] 2i + 5j. Then, find the angle between the two vectors.
Key Concepts
- The dot product of two vectors is the product of the i terms plus the product of the j terms.
- We can use the dot product to find the angle between two vectors.
- Dot products are useful for many types of physics applications.
Glossary
- dot product
- given two vectors, the sum of the product of the horizontal components and the product of the vertical components
Section 9.5 Homework Exercises
1. Given [latex]\boldsymbol{u}=\boldsymbol{i}−\boldsymbol{j}[/latex] and [latex]\boldsymbol{v}=\boldsymbol{i}+\boldsymbol{j}[/latex], calculate [latex]\boldsymbol{u}\cdot \boldsymbol{v}[/latex] and the angle between these two vectors.
2. Given [latex]\boldsymbol{u}=3\boldsymbol{i}−4\boldsymbol{j}[/latex] and [latex]\boldsymbol{v}=−2\boldsymbol{i}+3\boldsymbol{j}[/latex], calculate [latex]\boldsymbol{u}\cdot \boldsymbol{v}[/latex] and the angle between these two vectors.
3. Given [latex]\boldsymbol{u}=−\boldsymbol{i}−\boldsymbol{j}[/latex] and [latex]\boldsymbol{v}=\boldsymbol{i}+5\boldsymbol{j}[/latex], calculate [latex]\boldsymbol{u}\cdot \boldsymbol{v}[/latex] and the angle between these two vectors.
4. Given [latex]\boldsymbol{u}=\langle−2,4\rangle[/latex] and [latex]\boldsymbol{v}=\langle−3,1\rangle[/latex], calculate [latex]\boldsymbol{u}\cdot \boldsymbol{v}[/latex] and the angle between these two vectors.
5. Given [latex]\boldsymbol{u}=\langle−1,6\rangle[/latex] and [latex]\boldsymbol{v}=\langle 6,−1\rangle[/latex], calculate [latex]\boldsymbol{u}\cdot \boldsymbol{v}[/latex] and the angle between these two vectors.