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How do you find parametric equations and symmetric equations for the line through t0 and parallel to the given line t0 = (4, -2, 4) and x + 1 = y/2 = z + 5?

Answer 1

Emma Aldridge

#(x, y, z) = ( 4+t, -2+2t, 4+t)# that passes through #(4, -2, 4)# and is parallel to #x+1=y/2=z+5#.

The direction cosines of

#(x+1)/1=y/2=(z+5)/1#

are proportional to the denominators (1, 2, 1).

So, the line to this line is through #(4, -2, 4)# is given by

#(x-4)/1=(y+2)/2=(y-4)/1=t#,

where k is the parameter that gives the location of (x, y, z) on the

line, in the parametric form

#(x, y, z) = ( 4+t, -2+2t, 4+t)#

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Answer 2

Emily Ammerman

Parametric equations: [ x = 4 + 1t, \quad y = -2 + 2t, \quad z = 4 + 1t ]

Symmetric equations: [ \frac{x-4}{1} = \frac{y+2}{2} = \frac{z-4}{1} ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.