Question

Find sets of (a) parametric equations and (b) symmetric equations of the line through the two points. (For each line, write the direction numbers as integers.) $$ (0,0,25),(10,10,0) $$

Short answer

The parametric equations of the line are given by \(x = 10t\), \(y = 10t\), \(z = 25 - 25t\). The symmetric equations of the line are \(\frac{x}{10} = \frac{y}{10} = \frac{25 - z}{25}\).\nNote: This line is oriented along the direction vector from the first point to the second point.

Step-by-step solution

Identify the given points

The two points given in the problem are point A at the location \( (0, 0, 25) \) and point B at \( (10, 10, 0) \). We will use these points to find the direction vector.

Find the direction vector

The direction vector of a line passing through two points A and B can be found as \( \vec{AB} = \vec{OB} - \vec{OA} \). Here, \( \vec{OA} \) and \( \vec{OB} \) are position vectors of points A and B. For our points, the position vectors are \(\vec{OA} = \langle 0, 0, 25 \rangle\) and \(\vec{OB} = \langle 10, 10, 0 \rangle\). So, the direction vector \(\vec{AB}\) is \(\langle 10, 10, -25 \rangle \). This gives us the direction numbers of the line: 10, 10, and -25.

Formulate the parametric equations of the line

The parametric equations of the line can be written using the coordinates of any point on the line (we will use point A) and the direction numbers. Thus, for the direction numbers 10, 10, and -25 and point A at (0,0,25), our equations will be:\( x = 0 + 10t = 10t \) \\( y = 0 + 10t = 10t \) \\( z = 25 - 25t \)

Write the symmetric equations of the line

The symmetric equations of the line can be found by isolating the parameter 't' in each of the above parametric equations and equating them. This gives us:\(t = \frac{x}{10} = \frac{y}{10} = \frac{25 - z}{25} \)