Prove the identity u × (v × w) = (u · w)v − (u · v)w using the geometric definition of the cross product, by following the steps below. (a) First, prove it assuming v and w are parallel. (b) Assume v and w are not parallel. Argue that u × (v × w) lives in the plane containing the vectors v and w. Conclude that u × (v × w) = av + bw for some scalars a, b. (c) Assume both u · v and u · w are nonzero. Let a = c(u · w) for some scalar c. Show that b = −c(u · v). (Hint: Take the dot product of u with the both sides of the equation in (ii).) (Note: If one of them is zero, this is still true, but you don’t need to show it.)
Prove the identity
u × (v × w) = (u · w)v − (u · v)w
using the geometric definition of the cross product, by following the steps below.
(a) First, prove it assuming v and w are parallel.
(b) Assume v and w are not parallel. Argue that u × (v × w) lives in the plane containing the
(c) Assume both u · v and u · w are nonzero. Let a = c(u · w) for some scalar c. Show that
b = −c(u · v). (Hint: Take the dot product of u with the both sides of the equation in (ii).)
(Note: If one of them is zero, this is still true, but you don’t need to show it.)
(d) Consider u = i, v = i, w = j. What is the value of c? Conclude that
u × (v × w) = (u · w)v − (u · v)w.
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