Find a scalar equation of the plane that contains the given point P=(-3,-3,1)and the given normal vector n = (-1,4,7)
To find a scalar equation of a plane given a point and a normal vector, we can use the equation of a plane:
Ax + By + Cz = D
where A, B, and C are the coordinates of the normal vector (n) and D is the constant term.
In this case, we are given the point P=(-3,-3,1) and the normal vector n=(-1,4,7).
Step 1: Determine the values of A, B, and C.
The values of A, B, and C are the coordinates of the normal vector n. Therefore, A = -1, B = 4, and C = 7.
Step 2: Find the value of D.
To find the value of D, we substitute the coordinates of point P into the equation of the plane:
(-1)(-3) + (4)(-3) + (7)(1) = D
3 - 12 + 7 = D
-9 + 7 = D
D = -2
Step 3: Write the scalar equation of the plane.
Putting it all together, the scalar equation of the plane that contains the point P=(-3,-3,1) and has the normal vector n=(-1,4,7) is:
-1x + 4y + 7z = -2