Section 12.3 – The Dot Product
Exercises
Directions: You should try to solve each problem first, and then click "Reveal Answer" to check your answer. You can click "Watch Video" if you need help with a problem.
1. Compute \(\mathbf{a} \cdot \mathbf{b}\) if \(\mathbf{a} = \langle 11, -2, -1 \rangle\) and \(\mathbf{b} =\left< 3, 4, -7 \right>.\)
\(\mathbf{a} \cdot \mathbf{b} = 32\)
2. Compute \(\mathbf{a} \cdot \mathbf{b}\) if \(|\mathbf{a}| = 5\), \(|\mathbf{b}| = 2\), and \(\theta = 30^\circ.\)
\(\mathbf{a} \cdot \mathbf{b} = 5\sqrt{3}\)
3. Find the angle between the vectors \(\mathbf{a} = \langle 4, 2, -1 \rangle\) and \(\mathbf{b} = \left< -3, 0, 4 \right>.\)
\(\theta = \cos^{-1}\left(\dfrac{-16}{5\sqrt{21}}\right)\)
4. The points \(A(5, 1, 7)\), \(B(1, -2, 3)\), and \(C(2,-1, 6)\) form a triangle. Find \(\angle ABC.\)
\(\theta = \cos^{-1}\left(\dfrac{19}{\sqrt{41}\sqrt{11}}\right)\)
5. Find the scalar and vector projections of \(\left< 2, 4, 6 \right>\) onto \(\left< 1, 3, 5 \right>.\)
\(\textrm{comp}_{\mathbf{a}}\mathbf{b} = \dfrac{44}{\sqrt{35}},\) \(\quad \textrm{proj}_{\mathbf{a}}\mathbf{b} = \left< \dfrac{44}{35}, \dfrac{132}{35}, \dfrac{44}{7}\right>\)