# Determine the Cartesian equation of a plane passing through the points A (3,0,1) and B(0,1,-1) and perpendicular to the plane x- y- z + 1 = 0

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to write the Cartesian equation of the plane passing throught the given points and orthogonal to the plane `x- y- z + 1 = 0.`

You need to identify the normal vector to the plane `x- y- z + 1 = 0`  such that:

`bar n lt1,-1,-1gt`

You need to consider the points A(3,0,1) and B(0,1,-1) and the normal vector to write the cartesian equation of the plane:

`(y_B - y_A + z_B - z_A)(x - 1) + (x_B - x_A + z_B - z_A)(y + 1) + (x_B - x_A + y_B - y_A)(z + 1) = 0`

`(0 - 3 - 1 - 1)(x - 1) + (1 - 0 - 1 - 1)(y + 1) + ( 0 -3 + 1 - 0)(z+1) = 0`

`-5(x - 1) - (y + 1) - 2(z + 1) = 0`

`-5x + 5 - y - 1 - 2z - 2 = 0`

`-5x - y - 2z + 2 = 0`

Hence, evaluating the Cartesian equation of the plane under given conditions yields `-5x - y - 2z + 2 = ` 0.

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