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Exercises Trigonometry

Example 15 from Algebra and Trigonometry, 10.8 Vectors

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.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
Finding the dot product, we multiply corresponding components.
$\begin{array}{l} v_{1} \cdot v_{2} = 〈 5, 2 〉 \cdot 〈 3, 7 〉 \\ = 5 \cdot 3 + 2 \cdot 7 \\ = 15 + 14 \\ = 29 \end{array}$
To find the angle between them, we use the formula $\cos θ = \frac{v}{| v |} \cdot \frac{u}{| u |} .$
$\begin{array}{l} \frac{v}{| v |} \cdot \frac{u}{| u |} = 〈 \frac{5}{\sqrt{29}} + \frac{2}{\sqrt{29}} 〉 \cdot 〈 \frac{3}{\sqrt{58}} + \frac{7}{\sqrt{58}} 〉 \\ = \frac{5}{\sqrt{29}} \cdot \frac{3}{\sqrt{58}} + \frac{2}{\sqrt{29}} \cdot \frac{7}{\sqrt{58}} \\ = \frac{15}{\sqrt{1682}} + \frac{14}{\sqrt{1682}} = \frac{29}{\sqrt{1682}} \\ \begin{array}{l} = 0.707107 \\ \cos^{- 1} (0.707107) = 45° \end{array} \end{array}$
See Figure 17.

Figure 17

How did it go?

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From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 10.8, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.