# Find a vector that exists both in Col(A) and Col(B)

I this question from an old exam:

Determine all vectors located in Cola) and Col (B). Explain why all the vectors contained in both Col (A) and Col (B) form a subspace iR3 and calculate its dimension.

A below:

B below:

The thing is i understand the calculations but what i don't understand is that they take the cross-product of the vectors in Matrix A and same thing in Matrix B. And after that they take Cross-product of the both vectors and get a another 90 degrees vector and say this vector is both Col(A) and Col(B) how come that the Cross-Product can give a vectors in both spaces i don't understand i am very confused normally you use the Cross-Product to calculate a Equation for a plane?

Solution below:

Why can cross-product solve this problem i don't understand i am very confused ??

• The cross product does give you a vector in both spaces, but you are correct that that vector will not necessarily span $R(A) \cap R(B)$. For example, if $A = B = \begin{bmatrix} e_1 & e_2 \end{bmatrix}$, then you get $e_3 \times e_3 = 0$. Aug 12 at 18:38

Two 2-dimensional subspaces of $$\mathbb{R}^3$$ are either identical or have a 1-dimensional intersection. Clearly these two subspaces aren't identical, so we just need to find one vector in there intersection to find them all.

$$v\times w, v, w$$ forms a basis of $$\mathbb{R}^3$$ whenever $$v, w$$ are linearly independent. Furthermore, any vector orthogonal to $$v\times w$$ lies in the span of $$v, w.$$

So, $$(v\times w) \times (v'\times w')$$ will lie in the intersection of the subspace spanned by $$v, w$$ and the subspace spanned by $$v', w'.$$ Since the cross product is not 0 in this example, it must be spanning since the intersection in your case is 1-dimensional.

• So the Cross-Product gives a vector that is a bridge between two vectors?
– john
Aug 12 at 18:47
• @john What do you mean by bridge? A cross product gives a vector perendicular to both vectors.
– user147556
Aug 12 at 18:47
• I mean Vector that goes through both vectors? I understand what you mean by Perendicular!
– john
Aug 12 at 18:49
• @john It doesn't really go through either vector in any meaningful sense. It is perpendicular to both. Try graphing $(1, 0, 0), (0, 1, 0),$ and their cross product $(0, 0, 1)$ to see what the general geometric configuration (up to scaling and rotation) looks like.
– user147556
Aug 12 at 18:57
• Is the intersection in the subspaces then?
– john
Aug 12 at 19:00