Chapter 9.6, Problem 1E

A line in space is described algebraically by using __________ equations. The line that passes through the point P(x₀, y₀, z₀) and is parallel to the vector v = 〈a, b, c〉 is described by the equations x = __________, y = __________, z = __________.

To determine

To fill: A line in space is described algebraically by using _______ equations. The line passes through the point $P\left(x_0,y_0,z_0\right)$ and is parallel to the vector $v=\langle a,b,c\rangle$ is described by the equations x = _____, y = ______, z = ________.

Answer to Problem 1E

A line in space is described algebraically by using parametric equations. The line passes through the point $P\left(x_0,y_0,z_0\right)$ and is parallel to the vector $v=\langle a,b,c\rangle$ is described by the equations $x=x_0+at$, $y=y_0+bt$ and $z=z_0+ct$.

Explanation of Solution

The coordinates of the line are represented as functions of the same variable, they are called the parametric equations.

Thus, the equations which are used to describe the line in space algebraically is called parametric equations.

The parametric equations for the line which passes through the point $P\left(x_0,y_0,z_0\right)$ and parallel to the vector $v=\langle a,b,c\rangle$ is given below.

$x=x_0+at$ (1)

$y=y_0+bt$ (2)

$z=z_0+ct$ (3)

Thus, the line that passes through the point $P\left(x_0,y_0,z_0\right)$ and parallel to the vector $v=\langle a,b,c\rangle$ is described by the equations $x=x_0+at$, $y=y_0+bt$ and $z=z_0+ct$.