Lesson Video: Intersection of Planes Mathematics

Video Transcript

In this video, we will learn how to find points and lines of intersection between lines and planes in 3D space.

Recall that a plane in 3D space 𝑅 three may be described by the general equation 𝑎𝑥 plus 𝑏𝑦 plus 𝑐𝑧 plus 𝑑 equals zero, where 𝑎, 𝑏, 𝑐, and 𝑑 are all constants. Such a plane may look something like this on a three-dimensional graph. The plane consists of the coordinates of all possible points 𝑥, 𝑦, 𝑧 that satisfy the equation 𝑎𝑥 plus 𝑏𝑦 plus 𝑐𝑧 plus 𝑑 equals zero. Similarly to lines in 2D space, planes in 3D space extends infinitely in all directions. Assuming the plane is not parallel to any of the coordinate planes, that is, the 𝑥𝑦-, the 𝑦𝑧-, or the 𝑥𝑧-plane, for any given value of one variable, we can always find values for the other two variables that will solve the equation of the plane.

A plane’s orientation in space is defined by the coefficients of the variables in its equation: 𝑎, 𝑏, and 𝑐. More precisely, the coefficients define a vector 𝐧 that is normal to the plane and may be given by simply reading off the coefficients of the equation of the plane. 𝐧 is equal to 𝑎, 𝑏, 𝑐. Any nonzero scalar multiple of this vector, 𝛼𝐧, is also normal to the plane. So, for example, if we have the plane two 𝑥 plus three 𝑦 plus four 𝑧 minus five equals zero, we can obtain one normal vector 𝐧 by just reading off the coefficients of the variables in the equation. So 𝐧 is equal to two, three, four. But any scalar multiple of 𝐧 is also normal to the plane, for example, two 𝐧 equal to four, six, eight or negative 𝐧 equal to negative two, negative three, negative four.

If we have two different planes with normal vectors 𝐧 one and 𝐧 two, respectively, where 𝐧 two is a scalar multiple of 𝐧 one, then these two planes are parallel. Any two planes that do not have parallel normal vectors will intersect over a line in 3D space. Recall that in 2D space two nonparallel lines will intersect at exactly one point 𝑥 naught, 𝑦 naught. This point is the unique solution to the two equations of the two lines. We have the same number of equations as we do unknowns and therefore one unique solution.

In 3D space, things are a little different. Two nonparallel planes will intersect over a straight line in 3D space. This line contains the infinite number of points 𝑥, 𝑦, and 𝑧 that solve the two equations of the two planes. We have one more unknown than we do equations. Therefore, there are infinitely many solutions to these two equations which form the line in 3D space. This line is one-dimensional and may therefore be parameterized by one parameter. Let’s look at an example of how to find the general equation of a line between two planes.

Find the general equation of the line of intersection between the planes 𝑥 minus four 𝑦 plus three 𝑧 minus four equals zero and two 𝑥 plus two 𝑦 minus nine 𝑧 plus seven equals zero.

Let’s proceed as we normally would when solving two simultaneous equations by trying to eliminate one of the variables. We can eliminate 𝑧 by taking three times the first equation and adding it to the second equation. This gives five 𝑥 minus 10𝑦 minus five equals zero. Adding 10𝑦 plus five to both sides and dividing by five gives 𝑥 purely in terms of 𝑦: 𝑥 equals two 𝑦 plus one. Let’s move this equation up here for safekeeping. We can now go back to the equations of the planes and eliminate one of the other variables. We can eliminate 𝑦 by taking the first equation and adding two times the second equation. This gives five 𝑥 minus 15𝑧 plus 10 equals zero. Adding 15𝑧 and subtracting 10 from both sides and then dividing by five gives 𝑥 purely in terms of 𝑧: 𝑥 equals three 𝑧 minus two.

We now have two expressions for 𝑥, one in terms of 𝑦 and one in terms of 𝑧. Since both of these expressions are equal to 𝑥, they are also equal to each other. We can therefore rewrite these two equations as one equation with two equalities: 𝑥 equals two 𝑦 plus one equals three 𝑧 minus two. This is the general equation of the line of intersection between the two planes.

We cannot reduce the system of equations any further than this or find values for 𝑥, 𝑦, and 𝑧 that uniquely solve the equations because we have one more unknown than the number of equations and therefore an infinite number of solutions. However, we are free to set the value of one variable, which will give us corresponding values of the other two variables. This will give us one of the infinitely many solutions to the two equations, which is one point on the line of intersection.

For example, we can set 𝑧 equal to one. From this second part of the equation then, we have two 𝑦 plus one equals one. Rearranging for 𝑦 gives 𝑦 equals zero. From the other part of the equation, 𝑥 equals two 𝑦 plus one. And since 𝑦 is equal to zero, 𝑥 is equal to one. Therefore, 𝑥 equals one, 𝑦 equals zero, and 𝑧 equals one is one of the infinitely many solutions to the two equations of the two planes. And the point one, zero, one lies on the line of intersection.

The general equation is not the only way of describing the line of intersection between two planes. Another way is with a set of parametric equations, where 𝑥, 𝑦, and 𝑧 are each defined separately by an external parameter. Let’s look at an example of how to do this.

Find the parametric equations of the line of intersection between the two planes 𝑥 plus 𝑧 equals three and two 𝑥 minus 𝑦 minus 𝑧 equals negative two.

Recall that a straight line in 3D space may be described by the set of parametric equations 𝑥 equals 𝑥 naught plus 𝑎𝑡, 𝑦 equals 𝑦 naught plus 𝑏𝑡, and 𝑧 equals 𝑧 naught plus 𝑐𝑡. 𝑥 naught, 𝑦 naught, 𝑧 naught can be any point on the line 𝑝 naught, and 𝑎, 𝑏, 𝑐 is a direction vector that is parallel to the line. These parametric equations are arbitrary because 𝑝 naught can be any point on the line that we choose, and 𝐝 is just one direction vector that is parallel to the line. Any nonzero scalar multiple of 𝐝, 𝛼𝐝, is also parallel to the line.

To find the set of parametric equations for the line of intersection, we set one of these arbitrary parameterizations for one of the variables then substitute it into the two equations of the planes then rearrange the resulting equations to find expressions for the other two variables, also in terms of the parameter. For our two planes, we have the equations 𝑥 plus 𝑧 equals three and two 𝑥 minus 𝑦 minus 𝑧 equals negative two. If we substitute in an arbitrary parameterization for 𝑥, we get 𝑥 naught plus 𝑎𝑡 plus 𝑧 equals three and two times 𝑥 naught plus 𝑎𝑡 minus 𝑦 minus 𝑧 equals negative two. We can rearrange the first equation to give 𝑧 purely in terms of the parameter: 𝑧 equals three minus 𝑥 naught minus 𝑎𝑡.

Remember that 𝑥 naught and 𝑎 are both constants, so 𝑧 is a function of just the parameter 𝑡. We can now substitute this expression for 𝑧 into the second equation, which gives two times 𝑥 naught plus 𝑎𝑡 minus 𝑦 minus three minus 𝑥 naught minus 𝑎𝑡 equals negative two. Rearranging for 𝑦 gives 𝑦 purely in terms of the parameter: 𝑦 equals three times 𝑥 naught plus 𝑎𝑡 minus one. We now have 𝑥, 𝑦, and 𝑧 expressed purely as functions of the parameter 𝑡. So we have the arbitrary set of parametric equations. We are free to choose any values for 𝑥 naught and 𝑎 that we like with the exception that 𝑎 cannot be equal to zero because then changing the value of 𝑡 would not change the position on the line.

If we look at the list of possible answers, we can see that four of them have 𝑧 equals negative 𝑡. If we choose this as our parametric equation for 𝑧, this implies that three minus 𝑥 naught minus 𝑎𝑡 is equal to negative 𝑡. So this must mean that 𝑥 naught is equal to three and 𝑎 is equal to one. We can now substitute these values for 𝑥 naught and 𝑎 into the equations for 𝑥 and 𝑦. For 𝑥, we have 𝑥 equals three plus 𝑡, and for 𝑦 we have 𝑦 equals three times three plus 𝑡 minus one, which simplifies to eight plus three 𝑡. Our set of parametric equations therefore matches with answer (c) 𝑥 equals three plus 𝑡, 𝑦 equals eight plus three 𝑡, and 𝑧 equals negative 𝑡.

As we saw in the previous question, a line in 3D space may be defined by a point on the line and a direction vector parallel to the line. This hints at another way of describing a line in 3D space with a vector equation. Let’s look at an example of this.

Find the vector equation of the line of intersection between the planes 𝑥 plus three 𝑦 plus two 𝑧 minus six equals zero and two 𝑥 minus 𝑦 plus 𝑧 plus two equals zero.

Recall that the vector equation of a line in 3D space is given by 𝐫 equals 𝐫 naught plus 𝑡𝐝. 𝐫 naught is the position vector of a point on the line 𝑥 naught, 𝑦 naught, 𝑧 naught. 𝑡 is a scalar. And 𝐝 is a direction vector parallel to the line. This equation is not unique since we are free to choose any point that we like on the line for 𝐫 naught, and any vector that is a nonzero scalar multiple of 𝐝 will also be parallel to the line. So to find the vector equation for the line of intersection between the two planes, all we need to do is find a point in both planes with position vector 𝐫 naught and a direction vector 𝐝 that is parallel to the line of intersection.

Let’s begin by finding a point that lies in both planes for 𝐫 naught. Assuming the line of intersection is not parallel to any of the coordinate planes, we can choose any value for any one variable that we like and find corresponding values for the other two variables. This will give us one point that lies in both planes. Since every one of the possible answers has 𝑥 equal to zero in the constant vector, let’s try 𝑥 equals zero. If the line of intersection happens to be parallel to the 𝑦𝑧-plane, the value of 𝑥 will be constant and probably not equal to zero. In which case, setting 𝑥 equal to zero will mean that we will not be able to find solutions to the two equations. If this were the case, we would need to set some other variable equal to some other value.

Fortunately, that is not the case here. Setting 𝑥 equal to zero gives us three 𝑦 plus two 𝑧 minus six equals zero and negative 𝑦 plus 𝑧 plus two equals zero. We can simply rearrange the second equation to give 𝑦 equals 𝑧 plus two. And substituting this expression for 𝑦 into the first equation gives us three times 𝑧 plus two plus two 𝑧 minus six equals zero. And rearranging for 𝑧 gives us 𝑧 equals zero. And since 𝑦 is equal to 𝑧 plus two, 𝑦 is equal to two. So by setting 𝑥 equals zero into the two equations of the planes, we’ve obtained 𝑦 equals two and 𝑧 equals zero. So the point zero, two, zero lies in both planes. We therefore have the position vector 𝐫 naught of a point that lies on the line of intersection zero, two, zero.

We now need to find a direction vector 𝐝 that is parallel to the line of intersection. Since the line of intersection lies on both planes, its direction vector is parallel to both planes. If we look down the axis of the two planes intersecting, their line of intersection comes straight out of the screen. Their normal vectors, however, 𝐧 one and 𝐧 two, both lie in the plane of the screen. The cross product of the two normal vectors, 𝐧 one cross 𝐧 two, is perpendicular to both 𝐧 one and 𝐧 two and therefore also either come straight out of the screen or go straight into the screen. So we can use this as our direction vector for the line of intersection.

We can obtain the normal vectors 𝐧 one and 𝐧 two by simply reading off the coefficients of the variables in each of the equations of the planes. 𝐧 one is therefore equal to one, three, two and 𝐧 two is equal to two, negative one, one. Their cross product, 𝐧 one cross 𝐧 two, is given by the determinant of the three-by-three matrix 𝐢, 𝐣, 𝐤 followed by the components of 𝐧 one one, three, two and then the components of 𝐧 two two, negative one, one, where 𝐢, 𝐣, and 𝐤 are the unit vectors in the 𝑥-, 𝑦-, and 𝑧-directions, respectively. Expanding the determinant along the top row gives us 𝐢 times three minus negative two minus 𝐣 times one minus four plus 𝐤 times negative one minus six. Simplifying gives us five 𝐢 plus three 𝐣 minus seven 𝐤, which as a tuple is five, three, negative seven.

We therefore have our direction vector for the vector equation of the line of intersection: five, three, negative seven. Our complete vector equation for the line of intersection between the two planes is therefore 𝐫 equals zero, two, zero plus 𝑡 times five, three, negative seven, which matches with answer (d).

Sometimes instead of the line of intersection between two planes, we may wish to find the point of intersection between a line and a plane in 3D space. Let’s look at an example of this.

Find the point of intersection of the straight line negative three 𝑥 equals four 𝑦 minus two equals 𝑧 plus one and the plane negative three 𝑥 plus 𝑦 plus 𝑧 equals 13.

If they are not parallel or coplanar, a line and a plane in 3D space will intersect at a single point 𝑥 naught, 𝑦 naught, 𝑧 naught. This point is the unique solution to the equation of the line and the equation of the plane. We have three unknowns 𝑥, 𝑦, and 𝑧. We have one equation, the equation of the plane. And the equation of the straight line is effectively two equations since there are two equalities. We can rewrite the equation of the line as two distinct equations: negative three 𝑥 equals 𝑧 plus one, and four 𝑦 minus two equals 𝑧 plus one. We therefore effectively have three distinct equations for three unknowns. So as long as the line and the plane are not parallel or coplanar, there should be one unique solution to these three equations, which is the point where they intersect.

We can rewrite these two equations to give 𝑥 and 𝑦 both purely in terms of 𝑧. 𝑥 equals negative one-third times 𝑧 plus one, and 𝑦 equals one-quarter times 𝑧 plus three. We can then substitute these expressions for 𝑥 and 𝑦 into the equation of the plane, which will give us one equation for one variable 𝑧, which we can solve to find the value of 𝑧 and therefore the corresponding values of 𝑥 and 𝑦. So substituting in these expressions into the equation of the plane gives us negative three times negative one-third times 𝑧 plus one plus one-quarter times 𝑧 plus three plus 𝑧 equals 13.

Distributing the parentheses gives us 𝑧 plus one plus 𝑧 over four plus three-quarters plus 𝑧 equals 13. Rearranging by collecting terms in 𝑧 on the left-hand side gives us nine 𝑧 over four equals 45 over four. Multiplying by four and dividing by nine gives us 𝑧 equals five. We already have expressions for both 𝑥 and 𝑦 in terms of 𝑧, so we can substitute in this value of 𝑧 to give the values of 𝑥 and 𝑦. 𝑥 is equal to negative one-third times five plus one, which equals negative two, and 𝑦 is equal to one-quarter times five plus three, which is equal to two. The point of intersection between the line and the plane is therefore negative two, two, five.

Let’s finish this video by recapping some key points. Two nonparallel planes in 3D space will intersect over a straight line. This line contains the infinitely many points 𝑥, 𝑦, 𝑧 that satisfy the equations of the two planes. The line of intersection is one-dimensional. If it is not parallel to any of the coordinate planes, setting the value of one variable will give corresponding values for the other two. And finally, a line and a nonparallel plane in 3D space will intersect at a single point, which is the unique solution to the equation of the line and the equation of the plane.