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 the scalar equation of a plane that contains a given point and a given normal vector, we can follow these steps:
Step 1: Establish the equation of the plane in the general form
The general equation of a plane can be written as:
ax + by + cz + d = 0
Step 2: Substitute the given normal vector and point into the general equation
Using the point-normal form of the plane equation, we have:
(-1)(x + 3) + (4)(y + 3) + (7)(z - 1) = 0
Step 3: Simplify the equation
Distribute the coefficients:
-x - 3 + 4y + 12 + 7z - 7 = 0
Combine like terms:
-x + 4y + 7z + 2 = 0
Step 4: Rewrite the equation in scalar form
To convert the equation to scalar form, we can divide through by the negative coefficient of x:
x - 4y - 7z - 2 = 0
Therefore, the scalar equation of the plane that contains the point P=(-3, -3, 1) and the normal vector n=(-1, 4, 7) is:
x - 4y - 7z - 2 = 0.