Problem 1
(10 points)
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
(10 points)
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,...,vn be linearly independent. If a vector w can be written
w ,
then the choice of the coefficients α1,...,αn is unique.
Hint: Recall that a set of vectors is said to be linearly independent if w = 0 implies that all of the coefficients αk = 0 .
