Paul's Online Notes
Paul's Online Notes
Home / Calculus III / 3-Dimensional Space / Equations of Planes
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

Section 12.3 : Equations of Planes

4. Determine if the plane given by \(4x - 9y - z = 2\) and the plane given by \(x + 2y - 14z = - 6\) are parallel, orthogonal or neither.

Show All Steps Hide All Steps

Start Solution

Let’s start off this problem by noticing that the vector \({\vec n_1} = \left\langle {4, - 9, - 1} \right\rangle \) will be normal to the first plane and the vector \({\vec n_2} = \left\langle {1,2, - 14} \right\rangle \) will be normal to the second plane.

Now try to visualize the two planes and these normal vectors. What would the two planes look like if the two normal vectors were parallel to each other? What would the two planes look like if the two normal vectors were orthogonal to each other?

Show Step 2

If the two normal vectors are parallel to each other the two planes would also need to be parallel.

So, let’s take a quick look at the normal vectors. We can see that the first component of each vector have the same sign and the same can be said for the third component. However, the second component of each vector has opposite signs.

Therefore, there is no number that we can multiply to \({\vec n_1}\) that will keep the sign on the first and third component the same and simultaneously changing the sign on the second component. This in turn means the two vectors can’t possibly be scalar multiples and this further means they cannot be parallel.

If the two normal vectors can’t be parallel then the two planes are not parallel.

Show Step 3

Now, if the two normal vectors are orthogonal the two planes will also be orthogonal.

So, a quick dot product of the two normal vectors gives,

\[{\vec n_1}\centerdot {\vec n_2} = 0\]

The dot product is zero and so the two normal vectors are orthogonal. Therefore, the two planes are orthogonal.