anonymous 3 years ago Given $$\vec{g} = -3\vec{i} + 6\vec{j} - 12\vec{k}$$ and $$\vec{h} = 5\vec{i} - 10\vec{j} + 20\vec{k}$$, how could I prove that $$\vec{g}$$ and $$\vec{h}$$ are parallel? Please explain the reasoning.

1. anonymous

take their vector product, or prove they are multiple of each other

2. anonymous

Well does a vector product of 0 allow me to conclude that they are parallel?

3. anonymous

yes

4. anonymous

g x h=|g||h|sin(theta) where theta is angle between them. It will be = 0 only if angle is 0º or 180º

5. anonymous

Ok, just clarifying, is it because using the formal definition of the magnitude of a vector product, if $$\theta$$ = 0º or 180º, it would cause the entire thing to become 0, and therefore parallel since an angle of 0º or 180º would be the same line or parallel?

6. anonymous

yes

7. anonymous

Oh, never mind, thank you!

8. anonymous

yw

9. anonymous

more easy: -5/3(-3i+6j-12h)=5i-10j+20k

10. anonymous

so -5/3g=h

11. anonymous

Ok. that is the reason the book gave me, so I'm curious as to why that works...I don't see where -5/3 comes from.

12. anonymous

to make -3 equal to 5

13. anonymous

Oh. so you're multiplying $$\vec{g}$$ by -5/3 to make it equal $$\vec{h}$$?

14. anonymous

yes

15. anonymous

ok, once again, thank you :)

16. anonymous

yw, :)