# Problem 1

Prove the following identities for vectors *a,**b,**c *∈ R^{3}.

- The
*“BAC–CAB-identity”*

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

- 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 *∈ R^{3}.

- 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*).

- The identity k
*a*×*b*k^{2 }= k*a*k^{2 }k*b*k^{2 }− (*a**b*)^{2 }*.*

# Problem 3

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

*x** .*

- 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 *v*_{1}*,…,**v _{n }*be linearly independent. If a vector

*w*can be written

*w*,

then the choice of the coefficients *α*_{1}*,…,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 .

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