The Cross Product

1. Let \({\bf u} = \langle 2, 7, -5 \rangle \) and  \({\bf v } = \langle 1, 3, 2 \rangle \)  be given.  Find (i) \( {\bf u} \times {\bf v}\), and  (ii) \({\bf v} \times {\bf u} \).   solution

2. Use properties of cross product to find the following product s.    solution

(i) \( {\bf i} \times ({\bf j} \times {\bf k})\)   (ii) \(({\bf i} \times {\bf j}) \times ({\bf k} \times {\bf i}) \)   (iii)  \((3{\bf k} \times 4{\bf j}) \times {\bf j}\)   (iv) \( ({\bf j} + {\bf k}) \times ({\bf j} - {\bf k})

3.  Let \({\bf a} = \langle 1, 3, -2 \rangle \) and  \({\bf b } = \langle 0, -4, 3 \rangle \)  be given. Find two unit vectors orthogonal to both \( {\bf a }\) and \( {\bf b} \).   solution

4. Consider points A(3, −1, 2), B(2, 1, 5), and C(1, −2, −2). (a) Find the area of parallelogram ABCD with  adjacent sides \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\). (b) Find the distance from point A to line BC.   solution

5.Let \({\bf u} = \langle -3, 4, -1 \rangle \),   \({\bf  v } = \langle 0, -2, 3 \rangle \), and \({\bf w} = \langle 3,1,1\rangle \) be given. Find the volume of the parallelepiped with the adjacent edges \({\bf  u }, {\bf  v }\) and \({\bf  w }\).   solution