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.

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.

Show Solution

Finding the dot product, we multiply corresponding components.

[latex]\begin{align}{\boldsymbol{v}}_{1}\cdot {\boldsymbol{v}}_{2}&=\langle 5,2\rangle \cdot \langle 3,7\rangle \\ &=5\cdot 3+2\cdot 7 \\ &=15+14 \\ &=29 \end{align}[/latex]

To find the angle between them, we use the formula [latex]\cos \theta =\dfrac{\boldsymbol{v}}{|\boldsymbol{v}|}\cdot \dfrac{\boldsymbol{u}}{|\boldsymbol{u}|}[/latex].

[latex]\begin{align}\frac{\boldsymbol{v}}{|\boldsymbol{v}|}\cdot \frac{\boldsymbol{u}}{|\boldsymbol{u}|}&=\langle \frac{5}{\sqrt{29}}+\frac{2}{\sqrt{29}}\rangle \cdot \langle \frac{3}{\sqrt{58}}+\frac{7}{\sqrt{58}}\rangle \\ &=\frac{5}{\sqrt{29}}\cdot \frac{3}{\sqrt{58}}+\frac{2}{\sqrt{29}}\cdot \frac{7}{\sqrt{58}} \\ &=\frac{15}{\sqrt{1682}}+\frac{14}{\sqrt{1682}}=\frac{29}{\sqrt{1682}} \\ &=0.707107 \end{align}[/latex]

[latex]\theta={\cos }^{-1}\left(0.707107\right)=45^\circ[/latex]

Figure 1

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.