Section 12.4 – The Cross 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. If \(\mathbf{a} = \left< 3, -1, 7\right>\) and \(\mathbf{b} = \left< 2, 2, 5 \right>\), find \(\mathbf{b} \times \mathbf{a}.\)
\(\mathbf{b} \times \mathbf{a} = \left< 19,1,-8 \right>\)
2. Find a nonzero vector that is orthogonal to the plane passing through the points \(P(0, 1, 2)\), \(Q(4, -2, 5)\), and \(R(5, 4, -3).\)
\(\left< 6,35,27 \right>\) (or some scalar multiple of this, e.g., \(\left< -6,-35,-27 \right>)\)
3. Find the area of the parallelogram determined by \(\mathbf{a} = \left< 3, 0, 2 \right>\) and \(\mathbf{b} = \left< 1, -4, 5 \right>.\)
The area is \(\sqrt{377}.\)
4. Find \(|\mathbf{a} \times \mathbf{b}|\) if \(|\mathbf{a}| = 6\), \(|\mathbf{b}| = 2\), and \(\theta = 3\pi/4.\)
\(|\mathbf{a} \times \mathbf{b}| = 6\sqrt{2}\)