Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2013 / Paper 1 / 14 / c

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- Solution c

Solution

Use

(0, 1, 1)

as an anchor. Two directions in the required plane are

u = (2, 0, 2), v = (4, 1, 0) .

Their cross product is

(- 2, 8, 2)

, so a normal vector is

n = (- 1, 4, 1)

. The Cartesian equation of a plane is

n \cdot [(x, y, z) - (0, 1, 1)] = 0, - x + 4 y + z = 5.

Direct substitution verifies all three given points. The independent direction vectors ensure the plane is uniquely determined.

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