Math: Scalar Equations of a Plane

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Find the scalar equation of each of the following planes. State which of the planes, if any, are coincident.

a) r = (-8,-1,8) + s(-5,1,4) + t(3,2,-4)
Scalar Equation: -12x - 8y - 13z = 0

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b) r = (-2,-2,5) + s(3,1,-1) + t(4,1,-4)
Scalar Equation: -3x + 8y - z + 15 = 0

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*****How do you determine if the planes are the same?

• Math: Scalar Equations of a Plane -

(-5,1,4) and (3,2,-4) are two direction vectors on the first plane.
So we need a normal to these planes, the cross-product will give us that.

Which will be (12,8,13)
so the scalar equation will be
12x + 8y + 13z = D
(-8,-1,8) was given as a point on the plane, so
12(-8) + 8(-1) + 8(13) = D
D = 0

so the scalar equation is
12x + 8y + 13z = 0

Do the second one the same way.

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