Do the parametric equations #x = 3t^3 + 7#, #y = 2 - t^3#, #z = 5t^3 + 1# determine a line?

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Do the parametric equations #x = 3t^3 + 7#, #y = 2 - t^3#, #z = 5t^3 + 1# determine a line?

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Answer 1 · 2016-06-23T19:46:10

yes

Explanation:

well, you can see that

#[x,y,z]^T = [7,2,1]^T + t^3 [3,-1,5]^T#

if we replace #t^3# with #lambda# , which we are perfectly entitled to do, we see that this is the more recognisable equation of a straight line in #\mathbf{R}^3#:

#[x,y,z]^T = [7,2,1]^T + lambda [3,-1,5]^T#

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Do the parametric equations #x = 3t^3 + 7#, #y = 2 - t^3#, #z = 5t^3 + 1# determine a line?

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Explanation:

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