Video Transcript
Find the general form of the
equation of the plane passing through the point four, negative one, one and parallel
to the plane five 𝑥 plus six 𝑦 minus seven 𝑧 is equal to zero.
We begin by recalling that the
general form of the equation of a plane is written 𝑎𝑥 plus 𝑏𝑦 plus 𝑐𝑧 plus 𝑑
is equal to zero. This plane has normal vector 𝐧
equal to 𝑎, 𝑏, 𝑐. We also know that if two planes are
parallel, then their normal vectors must also be parallel. Our plane is parallel to the plane
five 𝑥 plus six 𝑦 minus seven 𝑧 equals zero, which has normal vector equal to
five, six, negative seven.
Any nonzero vector parallel to this
vector is a normal vector to the plane we want to write an equation of. This means that the simplest
parallel vector we can find is the exact same vector. This gives us the equation of our
plane five 𝑥 plus six 𝑦 minus seven 𝑧 plus 𝑑 equals zero. We know that this plane passes
through the point with coordinates four, negative one, one. We can therefore substitute in
these coordinates to calculate the value of 𝑑.
Five multiplied by four is 20, six
multiplied by negative one is negative six, and seven multiplied by one is seven,
giving us 20 plus negative six minus seven plus 𝑑 equals zero. This simplifies to seven plus 𝑑
equals zero. And subtracting seven from both
sides, we have 𝑑 is equal to negative seven. The general form of the equation of
the plane that passes through the point four, negative one, one and is parallel to
the plane five 𝑥 plus six 𝑦 minus seven 𝑧 equals zero is five 𝑥 plus six 𝑦
minus seven 𝑧 minus seven equals zero.
