Let L be the line through Po= (2, 1, 6), having direction vector u given by u = [4 -1 3]. (a) Find parametric equations for the line L. (b) Does the line L intersect the xy-plane? If so, what are the coordinates of the point of intersection?

Question

Let L be the line through [latex]P_{0}=(2,1,6)[/latex], having direction vector u given by

[latex]u =\left[\begin{array}{r}4 \-1 \3\end{array} ight][/latex].

(a) Find parametric equations for the line L.

(b) Does the line L intersect the xy-plane? If so, what are the coordinates of the point of intersection?

Accepted Answer

(a) One set of parametric equations for L is
x = 2 + 4t, y = 1 - t, z = 6 + 3t.

(b) If L intersects the x y- plane. it must do so at a point [latex]P_{1}[/latex] having z-coordinate equal to zero-a point of the form [latex] P_{1}=\left(x_{1}, y_{1}, 0\right)[/latex]. Since the z-coordinate of points on L is given by z = 6 + 3t, it follows that t = -2. When t = -2, we see that x = -6 and y = 3. Thus, the line L intersects the xy-plane at the point [latex]P_{1}=(-6,3,0)[/latex].

Question 2.4.1: Let L be the line through Po= (2, 1, 6), having direction ve...

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