Find P2 on Plane Given P1 & Dist. 3 Units

[Evien]

If you are given a plane, (x,y,z) = (2,3,0) + s(4,1,5) + t(1,2,6), and you have a point, let's say P1= (2,3,0). How would you find another point on the plane if you know the distance between P1 and the second point is 3 units?

 

[Studiot]

The locus of all points 3 units from a given one is a sphere.

This intersects the given plane in a circle.

So you need additional information to specify the second point you require.

 

[Evien]

Sorry forgot to mention that P1P2 is perpendicular to P1 and P3 (16.25, 0, 0).

 

[Studiot]

So do you not now have two unknowns, s and t, and two known points P1 and P3, that you can use to substitute into find s and t and thence P2?

 

[blue_raver22]

To find the second point, we can use the distance formula to set up an equation:

d = √[(x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2]

Where d is the distance between the two points, P1 = (x1, y1, z1) and P2 = (x2, y2, z2).

Since we know the distance between P1 and P2 is 3 units, we can substitute this value into the equation:

3 = √[(x2-2)^2 + (y2-3)^2 + (z2-0)^2]

Squaring both sides, we get:

9 = (x2-2)^2 + (y2-3)^2 + (z2)^2

Next, we can substitute the equation of the plane into the distance formula to eliminate one variable. This will give us a system of two equations:

x2 = 2 + 4s + t
y2 = 3 + s + 2t
z2 = 5s + 6t

Substituting these values into the equation above, we get:

9 = (2 + 4s + t - 2)^2 + (3 + s + 2t - 3)^2 + (5s + 6t)^2

Simplifying, we get:

9 = 16s^2 + t^2 + 20st + 4t + s^2 + 4st + 4t^2 + 25s^2 + 60st + 36t^2

Combining like terms, we get:

9 = 46s^2 + 84st + 41t^2

This is a quadratic equation in terms of s and t. We can solve for one variable and substitute the value into the equation of the plane to find the corresponding value for the other variable.

Once we have the values for s and t, we can plug them into the equation of the plane to find the coordinates of P2.

Therefore, using the given information and the distance formula, we can find the coordinates of P2 on the given plane.