Chapter 12 - Vectors and the Geometry of Space - 12.4 Exercises - Page 839: 22

$(a \times b) \cdot b= 0$

Let $a=\lt a_1,a_2,a_3\gt$ and $b=\lt b_1,b_2,b_3\gt$ $(a \times b) \cdot b=(\lt a_1,a_2,a_3\gt \times \lt b_1,b_2,b_3\gt) \cdot \lt b_1,b_2,b_3\gt$ $=\lt a_2b_3-a_3b_2,a_3b_1-a_1b_3,a_1b_2-a_2b_1 \gt \cdot \lt b_1,b_2,b_3\gt$ $=a_2b_1b_3-a_3b_1b_2+a_3b_1b_2-a_1b_2b_3+a_1b_2b_3-a_2b_1b_3$ $=0$ Hence, $(a \times b) \cdot b= 0$