Question

Let u = u1 i + u2 j + u3 k, v = v1 i + v2 j + v3 k, w = w1 i + w2 j + w3 k, Show that (u×v).w=(v×w).u=(w×u).v Using Cross Product properties.

Answer 1

The scalar triple products (u × v) · w, (v × w) · u, and (w × u) · v are equal due to the properties of the cross product and distributive property, demonstrated by rearranging and comparing the coefficients of the unit vectors.

Step-by-step explanation:

The question is asking to show that the scalar triple product (u × v) · w is equal to (v × w) · u, and also equal to (w × u) · v, using cross product properties. According to the cross product and distributive property, we can expand the products as follows:

• u × v = (u2v3 - u3v2)i + (u3v1 - u1v3)j + (u1v2 - u2v1)k
• v × w = (v2w3 - v3w2)i + (v3w1 - v1w3)j + (v1w2 - v2w1)k
• w × u = (w2u3 - w3u2)i + (w3u1 - w1u3)j + (w1u2 - w2u1)k

Then, we take the dot product with the remaining vector for each expression:

• (u × v) · w = (u2v3 - u3v2)w1 + (u3v1 - u1v3)w2 + (u1v2 - u2v1)w3,
• (v × w) · u = (v2w3 - v3w2)u1 + (v3w1 - v1w3)u2 + (v1w2 - v2w1)u3,
• (w × u) · v = (w2u3 - w3u2)v1 + (w3u1 - w1u3)v2 + (w1u2 - w2u1)v3.

By examining the coefficients of the unit vectors carefully, we see that the terms are rearranged, but they represent the same scalar quantity. Thus, showing that (u × v) · w = (v × w) · u = (w × u) · v.