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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.

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  1. myko
    • 2 years ago
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    take their vector product, or prove they are multiple of each other

  2. Calcmathlete
    • 2 years ago
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    Well does a vector product of 0 allow me to conclude that they are parallel?

  3. myko
    • 2 years ago
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    yes

  4. myko
    • 2 years ago
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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

  5. Calcmathlete
    • 2 years ago
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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?

  6. myko
    • 2 years ago
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    yes

  7. Calcmathlete
    • 2 years ago
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    Oh, never mind, thank you!

  8. myko
    • 2 years ago
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    yw

  9. myko
    • 2 years ago
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    more easy: -5/3(-3i+6j-12h)=5i-10j+20k

  10. myko
    • 2 years ago
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    so -5/3g=h

  11. Calcmathlete
    • 2 years ago
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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.

  12. myko
    • 2 years ago
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    to make -3 equal to 5

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

  14. myko
    • 2 years ago
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    yes

  15. Calcmathlete
    • 2 years ago
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    ok, once again, thank you :)

  16. myko
    • 2 years ago
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    yw, :)

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