3D Coordinate Geometry

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 $\frac{x-a}{p} = \frac{y-b}{q} = \frac{z-c}{r}.$

\frac{x-5}{4} = y-20 = \frac{z+1}{6}

\frac{x-4}{5} = \frac{y}{20} = -z+6

\frac{x+4}{5} = \frac{y}{20} = -z-6

\frac{x-4}{5} = \frac{y}{20} = z-6

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 $k?$

Concept Quizzes

Challenge Quizzes

3D Coordinate Geometry - Equation of a Line

Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, interactive explorations.

Used and loved by over 7 million people

Learn from a vibrant community of students and enthusiasts, including olympiad champions, researchers, and professionals.