CALC 3 : Is the line through (-4, -6, 1) and (-2, 0, -3) parallel to the line through (8, 19, 8) and (3, 4, 18)?

Question

Is the line through (−4, −6, 1) and (−2, 0, −3) parallel to the line through (8, 19, 8) and (3, 4, 18)?

 

Please use something along the lines of parametric and symmetric equations! This is section 12.5 Equations of Lines and Planes from James Stewart 8th edition. 

 

Thanks is advance!

Answer 1

Write a = (8 − 3)i + (19 − 4)j + (8 − 18)k or 5i + 15j − 10k.

Write b = (-2 − -4)i + (0 − -6)j + (-3 − 1)k or 2i + 6j − 4k.

By inspection, it is seen that 2i + 6j − 4k is (2/5) times 5i + 15j − 10k.

The line through (-4, -6, 1) and parallel to 5i + 15j − 10k is given by

[x,y,z] equal to [(-4 + 5t), (-6 + 15t), (1 − 10t)].

This last translates to t = (x + 4)/5 = (y + 6)/15 = (z − 1)/-10.

Place (x,y,z) equal to (-2,0,-3) into t = (x + 4)/5 = (y + 6)/15 = (z − 1)/-10

to obtain t = (-2 + 4)/5 = (0 + 6)/15 = (-3 − 1)/-10 or t = 2/5 = 2/5 = 2/5.

2/5 is the constant of proportionality seen above between vectors b & a.

It is proven in Calculus that these proportional vectors signify that the

lines specified above are parallel.