# Symmetry by plane

Determine the coordinates of a image of point A (3, -4, -6) at a symmetry that is determined by the plane x-y-4z-13 = 0

Correct result:

x =  -1
y =  1
z =  4

#### Solution:

r:x-y-4z-13=0 \ \\ A(3,-4,-6) \ \\ \ \\ n=s \ \\ n=(1,-1,-4) \ \\ p: x=t+a; y=-t+b; z=-4t + c; t \in R \ \\ \ \\ <\div class='alert alert-danger'>System of equations are non-linear.</\div> <\div class='alert alert-success'>The equations have the following integer solutions:</\div> <b>3=t + a \ \\ -4=-t + b \ \\ -6=-4 \cdot \ t + c \ \\ t=1</b><b>Number of solutions found: 1</b><h_{5}>a_{ 1</sub\geq_{2}, b_{ 1</sub\geq-3, c_{ 1</sub\geq-2, t_{ 1</sub\geq_{1}</h_{5}> \ \\ \ \\ p: x=t+2; y=-t-3; z=-4t-2; t \in R \ \\ \ \\ p(t=-1)=A' \ \\ \ \\ x=-1+a=-1
$y=-(-1)+b=1$
$z=-4 \cdot \ (-1)+c=4$

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Tips to related online calculators
For Basic calculations in analytic geometry is helpful line slope calculator. From coordinates of two points in the plane it calculate slope, normal and parametric line equation(s), slope, directional angle, direction vector, the length of segment, intersections the coordinate axes etc.

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