Question Video: Finding the Point of Intersection of Three Planes Mathematics

Video Transcript

Find the point of intersection of the planes negative five 𝑥 minus two 𝑦 plus six 𝑧 minus one equals zero, negative seven 𝑥 plus eight 𝑦 plus 𝑧 minus six equals zero, and 𝑥 minus three 𝑦 plus three 𝑧 plus 11 equals zero.

The three equations describe three planes in three-dimensional space. In general, though not always, three planes in space will intersect in a single point. In this case, they do. This point has coordinates 𝑥, 𝑦, and 𝑧. It is these coordinates that we need to find.

The coordinates of the point of intersection of the planes are the values of 𝑥 , 𝑦, and 𝑧 which simultaneously satisfy all three planar equations. Therefore, to find these coordinates, we need to solve this system of equations. First of all, we rearrange so that the constant terms are on the right-hand side. One way to solve this system of equations is to convert it into a single matrix equation and apply Cramer’s rule.

Cramer’s rule tells us that the solution to this matrix equations is given by the quotients of certain determinants. The determinant 𝛥 is the determinant of the coefficient matrix on the left. The determinant 𝛥 sub 𝑥 is the determinant of the matrix obtained by replacing the first column of the coefficient matrix with the one-by-three matrix on the right-hand side. The determinant 𝛥 sub 𝑦 is the determinant of the matrix obtained by replacing the second column with the matrix one, six, minus 11. Finally, 𝛥 sub 𝑧 is calculated by replacing the third column.

Let’s calculate these determinants. First, 𝛥 is the determinant of the coefficient matrix. This is equal to negative five times eight times three minus one times negative three minus negative two times negative seven times three minus one times one plus six times negative seven times negative three minus eight times one, which equals negative five times 27 plus two times negative 22 plus six times 13, which is negative 101.

Now let’s calculate 𝛥 sub 𝑥. 𝛥 sub 𝑥 equals one times eight times three minus one times negative three minus negative two times six times three minus one times negative 11 plus six times six times negative three minus eight times negative 11, which equals 505. 𝛥 sub 𝑦 comes out as 303. And 𝛥 sub 𝑧 is 505.

We can now substitute these values into Cramer’s rule. We have 𝑥 equals 505 divided by negative 101, which is negative five. 𝑦 equals 303 divided by negative 101, which is negative three. And 𝑧 equals 505 divided by negative 101, which is negative five. As discussed at the beginning, these are the coordinates of our point of intersection. The point of intersection is therefore negative five, negative three, negative five.