Learn / Library / Linear algebra

Onto a plane

Linear algebraMedium

Let $P$ be the $3 \times 3$ matrix of the orthogonal projection onto the plane $x + 2 y + 2 z = 0$ . What is the entry in row $1$ , column $1$ of $P$ ? Give a fraction in lowest terms.

Checking an answer needs JavaScript. The hint and the solution open without it.

Hint

Projecting onto a plane means removing the component along its normal.

Solution

With normal $n = (1, 2, 2)$ , the projection onto the plane is $P = I - \frac{n n ^{T}}{n ^{T} n}$ .
Here $n^{T} n = 1 + 4 + 4 = 9$ .
The entry in row $1$ , column $1$ of $n n^{T}$ is $1 \times 1 = 1$ .
So that entry of $P$ is $1 - \frac{1}{9} = \frac{8}{9}$ .

Answer: $\frac{8}{9}$

Next question Back to the library