Video Transcript
Determine whether the planes two,
three, negative two dot 𝐫 equals 12 and 𝑥 plus two 𝑦 plus four 𝑧 equals six are
parallel or perpendicular.
We begin by noting that the two
planes are given in different forms. The first plane, which we will call
𝑝 sub one, is written in vector form and has normal vector 𝐧 sub one equal to two,
three, negative two. The second plane 𝑝 sub two is
almost written in general form. The general form of a plane is 𝑎𝑥
plus 𝑏𝑦 plus 𝑐𝑧 plus 𝑑 equals zero, and this has normal vector 𝐧 equal to 𝑎,
𝑏, 𝑐. Subtracting six from both sides of
the equation, we have 𝑥 plus two 𝑦 plus four 𝑧 minus six equals zero, which is
the equation of the plane in general form. This has normal vector 𝐧 sub two
equal to one, two, four.
In this question, we are asked to
consider whether the two planes are parallel or perpendicular. If the two planes are parallel,
there is a nonzero scalar 𝑘 such that 𝐧 sub one is equal to 𝑘 multiplied by 𝐧
sub two. And if the two planes are
perpendicular, the dot product of the normal of vectors 𝐧 sub one and 𝐧 sub two
equal zero. Let’s begin by considering whether
the two planes are parallel. If this is true, then two, three,
negative two is equal to 𝑘 multiplied by one, two, four.
When multiplying a vector by a
scalar, we simply multiply each of the components by that scalar. This means that two, three,
negative two would need to be equal to 𝑘, two 𝑘, four 𝑘. For two vectors to be equal, each
of their components must be equal. Equating the 𝑥-components, we have
two is equal to 𝑘 or 𝑘 is equal to two. Equating the 𝑦-components, we have
three is equal to two 𝑘. And since 𝑘 is equal to two, this
gives us three is equal to two multiplied by two. As three is not equal to four, the
two components are not equal and the planes are therefore not parallel. We can confirm this by looking at
the 𝑧-components. We have negative two is equal to
four 𝑘. This time, we have negative two is
equal to eight, which once again is not true.
Let’s now consider whether the
planes are parallel. To do this, we will take the dot
product of the two normal vectors. The dot product is equal to the sum
of the product of the corresponding components. In this case, we have two
multiplied by one plus three multiplied by two plus negative two multiplied by
four. This simplifies to two plus six
plus negative eight, which is equal to zero. We can therefore conclude that the
two planes are perpendicular.
