MHB Finding Points Parallel to a Plane

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The vector P = (1,2,-1) is parallel to the plane defined by the equation 7x + 2y + kz = 5, leading to the determination of the value of k. To find k, the normal vector of the plane, represented as <7, 2, k>, is used in the equation for points parallel to the plane. The calculation shows that substituting the point (1,2,-1) into the plane equation yields 11 - k = 5. Solving this equation results in k being equal to 6. The discussion emphasizes the relationship between a point and a plane's normal vector in determining parallelism.
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If the vector $P = (1,2,-1)$ is parallel to the plane $7x+2y+kz = 5$, then what's the value of $k$?
 
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Do you know how to find the general formula for points parallel to a plane?
 
Deveno said:
Do you know how to find the general formula for points parallel to a plane?

So we find another plane parallel to this in which the point $(1,2,-1)$ lies?

I think $\mathbf{n} \cdot (\mathbf{x-x_{0}}) = 0$ so

$<7,2,k><x-1,y-2,z+1> = 0 \\
\implies 7x+2y+kz+k = 11 \\

\therefore 11-k = 5 \\

\implies k = 6.$
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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