3D Coordinate Geometry

3D Coordinate Geometry - Equation of a Line


If the line represented by x22=y33=z125\frac{x-2}{2} = \frac{y-3}{3} = \frac{z-12}{5} passes through the point (4,(4, a,a, b)b), what is the value of a+ba+b?

A line l l is parallel to the planes 4x+y=0 -4x + y = 0 and x+5z=0. x + 5z = 0. Given that it passes through the point (4,0,6), (4, 0, 6), find the equation of l l in the simplest form of xap=ybq=zcr. \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: {x12=y2p=z4p,x2q=y42=z73qq2,x50=y11r10=zr2. \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.p+q+r.

The following is the equation of a line with direction vector d=(p, q, 10):\vec{d} =(p,\text{ }q,\text{ }10): 2xu4=y22=z25.\frac{2x-u}{4}=\frac{y-2}{2} =\frac{z-2}{5}. If the line passes through the point (4, 4, 7),(4,\text{ } 4,\text{ } 7), what is p+q+up+q+u?

If the two lines x42=y+33=z44 and x6=yk2=z3 \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?k?


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