Answer to Question #106690 in Analytic Geometry for Caylin

Question

Suppose u and v are non zero factors in 3-space, where u=(u1,u2,u3) and v=(v1,v2,v3).

Prove that u * v is perpendicular to both u and v by making use of the det product.

Please assist.

Answer 1

Solution:

Let

$$d=u\times v$$

The coordinates of the cross product of vectors are determinants composed of the coordinates of the original vectors.

$$d (u_2v_3-u_3v_2; -u_1v_3+u_3v_2; u_1v_2-u_2v_1)$$

Vectors are perpendicular when their dot product is zero. 

The dot product of vectors is the sum of the products of the corresponding coordinates of the vectors.

$$du=u_1u_2v_3-u_1u_3v_2-u_1u_2v_3+u_2u_3v_1+u_1u_3v_2-u_2u_3v_1=0$$

$$dv=u_2v_1v_2-u_3v_1v_2-u_1v_2v_3+u_3v_1v_2+u_1v_2v_3-u_2v_1v_3=0$$