MHB Finding Points Parallel to a Plane

AI Thread Summary
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.
Guest2
Messages
192
Reaction score
0
If the vector $P = (1,2,-1)$ is parallel to the plane $7x+2y+kz = 5$, then what's the value of $k$?
 
Mathematics news on Phys.org
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.$
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...

Similar threads

Replies
3
Views
1K
Replies
1
Views
1K
Replies
7
Views
4K
Replies
6
Views
21K
Replies
3
Views
2K
Replies
8
Views
2K
Back
Top