Problem 1

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

1. The BACCAB-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 CauchyBinet 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 ₁,,_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 _k = 0 .