Question

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.

Answer 1

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

Answer 2

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

Answer 3

yes

Answer 4

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

Answer 5

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?

Answer 6

yes

Answer 7

Oh, never mind, thank you!

Answer 8

yw

Answer 9

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

Answer 10

so -5/3g=h

Answer 11

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.

Answer 12

to make -3 equal to 5

Answer 13

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

Answer 14

yes

Answer 15

ok, once again, thank you :)

Answer 16

yw, :)