Answer to Question #301385 in Analytic Geometry for pianowoman

Question

3a. Consider the line L that passes through the point P₀(4, 2, -3) is parallel to the vector

⟶ [ 2

u = -1 <----Matrix

6]

Find a vector equation and the parametric equations of the line L.

b. Find the point of intersection of the line L with the xy-plane (z = 0).

Answer 1

3.

a.  The vector equation of the line containing $P_0(4, 2, -3)$ and parallel to $\vec u=\begin{bmatrix} 2\\ -1 \\ 6 \end{bmatrix}$ is

$$\vec r=\begin{bmatrix} 4+2t\\ 2-t \\ -3+6t \end{bmatrix}$$

The corresponding parametric equations are

$$x=4+2t, y=2-t, z=-3+6t$$

b.

We have the line L

$$\dfrac{x-4}{2}=\dfrac{y-2}{-1}=\dfrac{z+3}{6}$$

If $z=0$

$$\dfrac{x-4}{2}=\dfrac{y-2}{-1}=\dfrac{0+3}{6}$$

$$x=5, y=3/2$$

Point $(5, 3/2, 0)$ is the point of intersection of the line L with the xy-plane.