Problem 1

Prove the following identities for vectors a,b,c ∈ R³.

1. The “BAC–CAB-identity”

a × (b × c) = b(a · c) − c(a · b). (1)

2. The Jacobi identity in three dimensions

a × (b × c) + b × (c × a) + c × (a × b) = 0.

Problem 2

Prove the following identities for vectors a,b,c,d ∈ R³.

1. The Cauchy–Binet formula in three dimensions

(a × b) · (c × d) = (a · c)(b · d) − (a · d)(b · c).

Hint: Use the identity u · (v × w) = v · (w × u).

2. The identity ka × bk² = kak² kbk² − (a b)² .

Problem 3

1. Find the minimum distance between the point p = (2,4,6) and the line

x .

2. Express the equation for the plane that contains the point p and the line x in parametric form. Then proceed to find the vector normal to this plane.

Bonus

Prove the following statement: Let v₁,…,v_n be linearly independent. If a vector w can be written

w,

then the choice of the coefficients α₁,…,α_n is unique.

Hint: Recall that a set of vectors is said to be linearly independent if w = 0 implies that all of the coefficients α_k = 0 .