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?
(-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.
To determine if two planes are the same, we can compare their scalar equations. If the scalar equations of two planes are equivalent, then the planes are coincident, which means they are the same plane or they coincide with each other.
In the case of the given scalar equations, we can compare the coefficients of the variables (x, y, z) as well as the constant terms.
For example, in part (a), the scalar equation is -12x - 8y - 13z = 0. In part (b), the scalar equation is -3x + 8y - z + 15 = 0.
By comparing the coefficients, we can see that the coefficient of x in part (a) is -12, while in part (b), it is -3. These coefficients are not equal, so the planes defined by these equations are not the same.
Similarly, the coefficients of y and z in part (a) are -8 and -13, respectively, while in part (b), they are 8 and -1. Again, these coefficients are not equal, indicating that the planes are different.
Therefore, none of the planes given in parts (a) and (b) are coincident.