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## 3D Coordinate Geometry

Append a z-axis to the 2-dimensional plane and conquer the realm of 3-dimensional space.

# Equation of a Line

If the line represented by $\frac{x-2}{2} = \frac{y-3}{3} = \frac{z-12}{5}$ passes through the point $$(4,$$ $$a,$$ $$b)$$, what is the value of $$a+b$$?

A line $$l$$ is parallel to the planes $$-4x + y = 0$$ and $$x + 5z = 0.$$ Given that it passes through the point $$(4, 0, 6),$$ find the equation of $$l$$ in the simplest form of $\frac{x-a}{p} = \frac{y-b}{q} = \frac{z-c}{r}.$

The following are the equations of lines that are parallel to one another: $\begin{cases} \frac{x-1}{2} = \frac{y-2}{p} = \frac{z-4}{p} _, \\\\ \frac{x-2}{q} = \frac{y-4}{2} = \frac{z-7}{3q-q^2}_, \\\\ \frac{x}{50} = \frac{y}{11 r - 10}=\frac{z}{r^2} _. \end{cases}$ Find the value of $$p+q+r.$$

The following is the equation of a line with direction vector $$\vec{d} =(p,\text{ }q,\text{ }10):$$ $\frac{2x-u}{4}=\frac{y-2}{2} =\frac{z-2}{5}.$ If the line passes through the point $$(4,\text{ } 4,\text{ } 7),$$ what is $$p+q+u$$?

If the two lines $\frac{x-4}{2} = \frac{y+3}{3} = \frac{z-4}{4} \text{ and } x-6 = \frac{y-k}{2} =z-3$ intersect, what is $$k?$$

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