Description
Problem 1
(10 points)
Prove the following identities for vectors a, b, c ∈ R
3
.
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 ∈ R
3
.
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 × bk
2 = kak
2
kbk
2 − (a · b)
2
.
Problem 3
(10 points)
1. Find the minimum distance between the point p = (2, 4, 6) and the line
x =
−1
1
6
+ λ
1
−1
0
.
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
(10 points)
Prove the following statement: Let v1, . . . , vn be linearly independent. If a vector w can be
written
w =
Xn
k=1
αk vk ,
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 .
