Find a set of symmetric equations of the line that passes through the points $( 5,0,5 )$ and $( 7,8 , - 6 )$.
A) $\frac { x } { 7 } = \frac { y } { 8 } = \frac { z } { - 6 }$
B) $\frac { x - 5 } { 2 } = \frac { y - 8 } { 8 } = \frac { z - 6 } { - 11 }$
C) $\frac { x - 5 } { - 2 } = \frac { y } { 8 } = \frac { z - 5 } { 11 }$
D) $\frac { x + 2 } { 5 } = y - 8 = \frac { z - 11 } { 5 }$
E) $\frac { x - 5 } { 2 } = \frac { y } { 8 } = \frac { z - 5 } { - 11 }$

Question

Quizplus · Accepted Answer

We can use the formula for the symmetric equations of a line which is:
$$\frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}$$
where $(x_0,y_0,z_0)$ is a point on the line and $(a,b,c)$ is the direction vector of the line.

First, we need to find the direction vector $\vec{v}$ of the line passing through $(5,0,5)$ and $(7,8,-6)$. This is given by:
$$\vec{v}=(7,8,-6)-(5,0,5)=(2,8,-11)$$
Note that this vector is not normalized, but we don't need to normalize it because we just want the direction of the line.

Next, we can use either of the given points on the line as our $(x_0,y_0,z_0)$ in the formula for the symmetric equations. Let's use $(5,0,5)$:
$$\frac{x-5}{2}=\frac{y-0}{8}=\frac{z-5}{-11}$$
This is exactly what choice E gives us, so the answer is E.

Find a Set of Symmetric Equations of the Line That (5,0,5)( 5,0,5 )(5,0,5)

Find a Set of Symmetric Equations of the Line That $( 5,0,5 )$