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Append a z-axis to the 2-dimensional plane and conquer the realm of 3-dimensional space.
Determine the value of \( b\) such that the line \( \frac{x-1}{4} = \frac{y-2}{b} = \frac{z}{9} \) does not intersect the plane \( 2x- 4y + 5z = 6. \)
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If the angle between the line \( x= \frac{y-4}{2}=\frac{z-1}{\lambda} \) and the plane \( x + 3y + z = 1 \) is \( \cos ^{ -1 }{ \left( \sqrt { \frac { 10 }{ 11 } } \right) }, \) what is \( \lambda?\)
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Find the symmetric point of \( A = (4, 4, 16) \) to the plane \[ \pi : x+ y+z=21. \]
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Let \(\alpha,\beta,\) and \(\gamma\) be the angles that the line \(\frac{x-2}{3}=-\frac{y+1}{4}=\frac{z+3}{5}\) forms with the \(xy\)-, \(yz\)-, and \(xz\)-planes. Then what is the value of \(\sin^2\alpha+\sin^2\beta+\sin^2\gamma?\)
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What is the volume of the region bounded by the plane \(7x+8y+9z-504=0\) and the \(xy\)-, \(yz\)-, and \(xz\)-planes?
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