# Homework Help: Cross product question

1. Apr 20, 2010

### spoc21

1. The problem statement, all variables and given/known data

Prove that If $$\vec{a}$$x $$\vec{b}$$ = 0, then $$\vec{a}$$is parallel to $$\vec{b}$$.

2. Relevant equations

3. The attempt at a solution
I tried attempting the solution by using the following:

$$\vec{a}$$ = [a1, a2, a3]
$$\vec{b}$$ = [b1, b2, b3]

When I took the cross product of a x b I got::

[a2b3 - b2a3 a3b1 - b3a1, a1b2 - b1a2]

and we can make this equal to 0, but I am confused here; I have no idea on how to prove that when a x b is 0, vectors a and b are parallel.

any help is appreciated!

Last edited: Apr 20, 2010
2. Apr 20, 2010

### Wingeer

Imagine the zero as the zero vector, defined in R3 as 0=[0,0,0]

Edit: You could view the geometric interpretation. In which case you will have a parallellogram with area equal to zero.

Last edited: Apr 20, 2010
3. Apr 20, 2010

### spoc21

thanks, so this is what I end up with:

a2b3 = b2a3.....(1)

a3b1 = b3a1....(2)

a1b2 = b1a2....(3)

Does this suggest that the two vectors, a and b are parallel?

4. Apr 21, 2010

### spoc21

Bump..any one?

thanks

5. Apr 21, 2010

### Susanne217

Its it simply

Let $$\vec{a},\vec{b} \neq 0$$ and let

$$\vec{a} \times \vec{b} = |\vec{a}| \cdot |\vec{b}| \cdot sin(\theta}) \cdot \mathrm{n}$$ be the definition of the cross product where $$0 \leq \theta \leq \pi$$

then for $$\theta = 0$$ $$\vec{a} \times \vec{b} = |\vec{a}| \cdot |\vec{b}| \cdot sin(0}) \cdot \mathrm{n} = 0$$

q.e.d.

Last edited: Apr 21, 2010
6. Apr 23, 2010

### spoc21

7. Apr 23, 2010

### tiny-tim

Hi spoc21!
Yes, because they become

a2/a3 = b2/b3.....(1)

a3/a1 = b3/b1....(2)

a1/a2 = b1/b2....(3)

8. Apr 25, 2010

### spoc21

Thanks tiny-tim So basically if I write the vectors in this form, I am showing that they are parallel, and this would be enough for the proof? Also, just for my knowledge could you please elaborate on this..
Thank you very much

9. Apr 25, 2010

### tiny-tim

Hi spoc21!
Yes, that's enough to do it.

To elaborate …two vectors are parallel if one is a scalar times the other …

and you can easily check that that means that the ratios of their coordinates must be the same.

10. Apr 25, 2010

Thanks!!