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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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