Find the vector equation of the line through (00,00,00) and (11,55,44) where tequals=0 corresponds to the first given point and where tequals=1 corresponds to the second given point.

  (x, y, z) = (0, 0, 0) +t(1, 5, 4)

Step-by-step explanation:

Such a parametric equation can be written in the form ...

  (first point) + t×(change in point values)

where the change in point values is the difference between the point coordinates.

Since the first point is (0, 0, 0), the change is easy to find. It is exactly equal to the second point's coordinates. Hence our equation is ...

  (x, y, z) = (0, 0, 0) +t(1, 5, 4)

The sum can be "simplified" to ...

  (x, y, z) = (t, 5t, 4t)

Comment on answer forms

I like the first form for many uses, but the second form can preferred in some circumstances. Use the one consistent with your reference material (textbook or teacher preference).

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step-by-step explanation:

4  

25

5258y

54.

t5

5r

54

5856

542

5458

86

.8558.65

565

565

55898

786

8/95

5896

8656265

556478