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3D Coordinate Geometry

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

Perpendicular Planes

         

Which of the following planes is perpendicular to the plane that contains the points \[ (0,0,0), (3,2,2 ), (6,-3,-10)? \]

If the plane \( a x + b y + 3z = 1 \) is perpendicular to the planes \[\begin{align} -3 x + 4 y - 2 z &= 0 \\ 2 x - 3 y &= 5,
\end{align}\] what is the value of \( a + b ? \)

Which of the following planes is perpendicular to the plane \[ -2x + 3 y -5 z = 1? \]

Which of the following planes is perpendicular to the plane \[ 2x - 7 y + 6 z = 6? \]

\(\alpha, \beta \text{ and } \gamma\) are planes such that \(\alpha\) and \(\beta\) do not intersect and \(\alpha\) and \(\gamma\) are perpendicular. Given that lines \(l, m \text{ and } n\) lie on planes \(\alpha, \beta \text{ and } \gamma,\) respectively, which of the following statement(s) is(are) true?

I. \(\beta\) and \(\gamma\) are perpendicular.
II. There exist \(m\) and \(n\) that intersect.
III. If \(l \text{ and }m\) are perpendicular to \(\gamma\), then \( l // m.\)

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