Finding the point of intersection and distance of a vector

[InvaznGSC]

Homework Statement

Let P be the plane of Problem 9 and let L be the line perpendicular to P and passing through the point p=(1,1,2). Find the point of intersection between P and L and use it to compute the distance from p to P.

Question 9 is this:
Use the cross product to obtain the normal and standard equations of the plane P of problem 5.

Question 5 is this:
Verify that the points p_1=(0,0,1), p_2=(1,1,1) and p_3=(-1,0,1) are not aligned, and find the vector parametric equation of the plane P in R^3 that passes through them.

Homework Equations

Would I have to the the standard/normal equation from question 9 or the vector parametric equation from question 5?

How do you integrate the point, p=(1,1,2) into the any of the above equations?

Would the line perpendicular to P be similar to the normal found in question 9?

The Attempt at a Solution

For question 5:
q=P_1+(P_2-P_1)
q=(0,0,1)+t(1,1,0)
(-1,0,1) = (0,0,1)+t(1,1,0)

-1 = 0+1t --> -1=t
0=0+1t --> 0=t
1=1+0t --> 1=1

Therefore they are not aligned.

Vector parametric equation:
[x,y,z]=q=P_1+t(P_2-P_1)+s(P_3-P_1)
[x,y,z]=q=(0,0,1)+t(1,1,0)+s(-1,0,0)

For question 9:
P_1P_3 = (-1,0,0) P_1P_3 = (1,1,0)

I used the cross product to find the normal to be (0,0,-1) then plugged it into the equation:
a(x-x_0)+b(y-y_0)+c(z-z_0) = 0

and then i got:

0x+0y-z+1 = 0 as my standard equation

I honestly have no idea where to begin for problem 10. I tried placing the point (1,1,2) into both equations, and I'm not sure if I am supposed to do that, and if I was, I wouldn't know how to use it to find the distance. :/

 

[mighty2000]

Hello,

For problem 10, you can use the point of intersection between P and L to find the distance from p to P. To find this point of intersection, you can set the parametric equation of the line L equal to the standard equation of the plane P and solve for t and s. This will give you the coordinates of the point of intersection.

Once you have the coordinates of the point of intersection, you can use the distance formula to find the distance from p to P. This formula is: distance = √((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2), where (x1,y1,z1) are the coordinates of p and (x2,y2,z2) are the coordinates of the point of intersection.

Hope this helps!