Question

Prove that (a − b) ⨯ (a + b) = 2(a ⨯ b). The following properties and theorem may be used in your proof. Properties of the cross product: If a, b, and c are vectors and c is a scalar, then we have the following properties. a ⨯ b = −b ⨯ a (ca) ⨯ b = c(a ⨯ b) = a ⨯ (cb) a ⨯ (b + c) = a ⨯ b + a ⨯ c (a + b) ⨯ c = a ⨯ c + b ⨯ c a · (b ⨯ c) = (a ⨯ b) · c a ⨯ (b ⨯ c) = (a · c)b − (a · b)c Theorem 1: Two nonzero vectors a and b are parallel if and only if a ⨯ b = 𝟬. Complete the proof by following the justifications provided. (a − b) ⨯ (a + b) = by Property 3 = by Property 4 = by Property 2 (with c = −1) = by Theorem 1 = by Property 1 = 2(a ⨯ b)

          Prove that (a − b) ⨯ (a + b) = 2(a ⨯ b).
The following properties and theorem may be used in your proof.
Properties of the cross product:
If a, b, and c are vectors and c is a scalar, then we have the following properties.
    a ⨯ b = −b ⨯ a
    (ca) ⨯ b = c(a ⨯ b) = a ⨯ (cb)
    a ⨯ (b + c) = a ⨯ b + a ⨯ c
    (a + b) ⨯ c = a ⨯ c + b ⨯ c
    a · (b ⨯ c) = (a ⨯ b) · c
    a ⨯ (b ⨯ c) = (a · c)b − (a · b)c
Theorem 1:
Two nonzero vectors a and b are parallel if and only if
a ⨯ b = 𝟬.
Complete the proof by following the justifications provided.
(a − b) ⨯ (a + b)  =  
    by Property 3
  =  
  by Property 4
  =  
  by Property 2 (with c = −1)
  =  
  by Theorem 1
  =  
  by Property 1
  =  2(a ⨯ b)

Added by Yolanda M.

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