Vector identities

Question 1

What is ∇ • (F × G) equal to?

Answer 1

G • (∇ × F) – F • (∇ × G)

Question 2

What does it mean when dot product equals zero?

Answer 2

The components in question are orthogonal to each other.

Question 3

What are the properties of a linear vector space with dimension n; n ∈ N?

Answer 3

1. The basis of a vector space has n linearly independent vectors
2. n + 1–elements must be linearly dependent,
3. If U and W are subspaces of a vector space V, then,
   dim(u + w) = dim(u) + dim(w) – dim(u intersect w)

Question 4

What can be said about the determinant of a set of basis?

Answer 4

The determinant of the basis must be non zero, as basis are linearly independent.

Determinant must be gotten by

Question 5

What can be said about the determinant of vectors in checking whether or not they are linearly dependent?

Answer 5

If they are linearly dependent, the determinant will be zero. If they are linearly independent, the determinant will be non-zero.

Question 6

What can be said about the divergence of a rotor (curl) and the rotor of a gradient?

Answer 6

DORO GO:

The divergence of a rotor is zero, as is the rotor of a gradient.