Given the planes P1 : x + 4y − z = 10, P2 : 3x − y + 2z = 4, and the point A (−2, 1, 0).

(a) Determine an equation for the plane that contains A and is parallel to P1.
(b) Determine whether A lies on P2. If it doesn’t, find the distance from A to P2.

a) x+4y-z=2

b)   -7/

Step-by-step explanation:

a)

Equation of a plane containing point A is given by;

[x-(-2)]+[y-1]+[z-0]=0 ----(1)

If this plane is parallel to the plane P1 whose normal vector is {1.4.-1] - then this will also be normal to plane (1)

so,

(1)[x-(-2)]+(4)[y-1]+(-1)[z-0]=0

==> x+2+4y-4-z=0

==> x+4y-z=2

will be the plane parallel to P1

b) If A lies on P2 then it should satisfy the equation of P2

Putting x= -2, y=1 and z= 0

==> 3(-2)-(1)+2(0)=4

==>-6-1+0=4

==> 5=4

which is not true i-e equation is not satisfied ! So the point lies out of plane. Lets find distance from P2 of point A

We know distance of point (x1,y1,z1) from a plane Ax+By+CZ+D=0 is

d= Ax1+By1+Cz1+D /

So distance of point A from P2 is;

d= 3×(-2)-1(1)+2(0)/

==> d= -7/

answer: a video game has three difficulty levels: easy, medium, and expert. the game's scoring works in such a way that a player has to score a fixed number of points to advance to the next level, and then the scoring starts over. the total score needed to clear all three difficulty levels is 7,500. two times the number of points needed to clear the easy level is 500 more than the number of points needed to clear the medium level. twice the number of points needed to clear the medium difficulty level is 1,500 more than the number of points needed to clear the expert level. if x, y, and z represent the easy, medium, and expert levels, respectively, and a is the coefficient matrix of the system of equations modeling this situation, identify the inverse matrix, a-1, and the solution matrix, x, of the system of equations.

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