## Calcmathlete 2 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. myko

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

2. Calcmathlete

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

3. myko

yes

4. myko

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. Calcmathlete

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. myko

yes

7. Calcmathlete

Oh, never mind, thank you!

8. myko

yw

9. myko

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

10. myko

so -5/3g=h

11. Calcmathlete

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. myko

to make -3 equal to 5

13. Calcmathlete

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

14. myko

yes

15. Calcmathlete

ok, once again, thank you :)

16. myko

yw, :)