Calcmathlete
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.
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myko
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take their vector product, or prove they are multiple of each other
Calcmathlete
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Well does a vector product of 0 allow me to conclude that they are parallel?
myko
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yes
myko
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g x h=|g||h|sin(theta)
where theta is angle between them. It will be = 0 only if angle is 0º or 180º
Calcmathlete
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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?
myko
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yes
Calcmathlete
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Oh, never mind, thank you!
myko
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yw
myko
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more easy:
-5/3(-3i+6j-12h)=5i-10j+20k
myko
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so -5/3g=h
Calcmathlete
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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.
myko
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to make -3 equal to 5
Calcmathlete
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Oh. so you're multiplying \(\vec{g}\) by -5/3 to make it equal \(\vec{h}\)?
myko
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yes
Calcmathlete
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ok, once again, thank you :)
myko
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yw, :)