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Excel in math and science
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It's hard to learn from lectures and videos
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3D Coordinate Geometry
Append a z-axis to the 2-dimensional plane and conquer the realm of 3-dimensional space.
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, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Concept Quizzes
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 \(a+b\)?
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, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
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 \[ \frac{x-a}{p} = \frac{y-b}{q} = \frac{z-c}{r}. \]
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, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
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 \(p+q+r.\)
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, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
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 \(p+q+u\)?
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, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
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 \(k?\)