Because the given and the required planes are parallel their x, y and z coefficient denoted by A, B and C respectively are the same
so we have A = 2, B = 1 and C = −2
The new plane should pass through the point P0. And the equation of the required plane is:
A (x − x0) + B (y − y0) + C (z − z0) = 0
Substituting values of the point and the direction numbers we get the equation:
2 (x − 2) + 1 (y − 1) − 2(z + 1) = 0
After arranging terms, we get the final plane equation:
2x + y − 2z − 7 = 0
If the plane is given by Ax + By + Cz + D = 0
The parametric line equations connecting point P0 (x0 , y0 , z0) are
x = x0 + At |
y = y0 + Bt |
z = z0 + Ct |
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If the intersection point of the line and the plane is: P1 (x1 , y1 , z1)
The square distance from point P0 to point P1 is: d2 = (x0 − x1)2 + (y0 − y1)2 + (z0 − z1)2(1)
Because point P1 is on the line and also on the plane the line parametric equations are:
x1 = x0 + At1 |
y1 = y0 + Bt1 |
z1 = z0 + Ct1 |
(2) |
Point x1 is located on the plane hence it satisfies the plane equation:
Ax1 + By1 + Cz1 + D = 0
After substitute eq (2) into the plane equation we get:
A(x0 + At1) + B(y0 + Bt1) + C(z0 + Ct1) + D = 0(3)
Now substitute equation (2) into equation (1) we get
d2 = (At1)2 + (Bt1)2 + (Ct1)2 = t12 (A2 + B2 + C2)
After eliminating t1 we get: |
$$ t_1 = {d \over \sqrt{A^2+B^2+C^2}}$$ |
(4) |
Substituting the value of t1 from equation (4) into equation (3) we get:
$$ Ax_0 + {A^2d \over \sqrt{A^2+B^2+C^2}} + By_0 + {B^2d \over \sqrt{A^2+B^2+C^2}} + Cz_0 + {C^2d \over \sqrt{A^2+B^2+C^2}}+D=0 $$
After arranging terms and some algebraic steps we get the final value for d:
$$ {d \over \sqrt{A^2+B^2+C^2}} (A^2+B^2+C^2) +Ax_0+By_0+Cz_0+D=0$$
$$ d= {-(Ax_0+By_0+Cz_0+D)\sqrt{A^2+B^2+C^2} \over A^2+B^2+C^2}$$
$$ d={|Ax_0 + By_0 + Cz_0 + D| \over \sqrt{A^2+B^2+C^2}}$$
And after substituting values we get:
$$ d={|2‧2 + 1‧1 + 2‧1 + 5| \over \sqrt{4+1+4}} = {12\over \sqrt{9}} = {4}$$
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