Question

Find both the vector equation and the parametric equations of the line through (0 comma 0 comma negative 2 )(0,0,−2) in the direction of the vector Bold vvequals=left angle negative 2 comma negative 4 comma 0 right angle−2,−4,0​, where tequals=0 corresponds to the given point.

Answer 1

The line (-2, -4, 0)t points in the same direction as the given vector (-2, -4, 0) and passes through the origin. To make this line pass through the point (0, 0, -2), all we need to do is translate the line by the vector (0, 0, -2). So the line has vector equation

u(t) = (-2, -4, 0)t + (0, 0, -2) = (-2t, -4t, -2)

and we see that t = 0 indeed corresponds to the given point.

In parametric form, the line is given by

x(t) = -2t

y(t) = -4t

z(t) = -2

for any real number t.