Question Video: Finding the Vector Form of the Equation of a Plane Mathematics

Video Transcript

The equation of a plane has general form five 𝑥 plus six 𝑦 plus nine 𝑧 minus 28 is equal to zero. What is its vector form?

In this question, we’re given the general form of the equation of the plane and we need to use this to determine the vector form of the equation of the plane. To answer this question, let’s start by recalling what we mean by the general form of the equation of a plane. It’s the form 𝑎𝑥 plus 𝑏𝑦 plus 𝑐𝑧 plus 𝑑 is equal to zero, where 𝑎, 𝑏, 𝑐, and 𝑑 are constants. And in particular, the vector 𝐚, 𝐛, 𝐜 will be a normal vector to the plane. And since we’re given the general form of the equation of the plane, we can determine the normal vector to the plane as the coefficients of our variables. And we can also determine the value of 𝑑.

Let’s now recall the vector form of the equation of a plane. It’s the form vector 𝐧 dot vector 𝐫 is equal to a constant 𝑑 prime, where the vector 𝐧 is a normal vector to the plane. And there’s something worth noting about the constants 𝑑 prime and 𝑑. There’s a geometric interpretation of both of these constants. The constant 𝑑 prime is a translation of our plane 𝑑 prime units in the direction of the normal vector 𝐧. However, the constant 𝑑 is a translation of our plane negative 𝑑 units in the direction of its normal vector. So if we set our normal vector 𝐧 equal to the vector 𝐚, 𝐛, 𝐜, then we get that 𝑑 prime will be equal to negative 𝑑.

And it’s worth noting there is a second nongeometric way of seeing this result. We just subtract 𝑑 from both sides of the equation in our general equation of the plane. So we’ll set our vector 𝐧 to have components equal to the coefficients of the variables. That’s 𝐧 as the vector five, six, nine. And then our constant 𝑑 prime will be negative 𝑑. That’s negative one times negative 28, which is just 28. And then we substitute this vector and this constant into our vector form of the equation of a plane to get our final answer, which is that the vector form of the equation of the plane five 𝑥 plus six 𝑦 plus nine 𝑧 minus 28 is equal to zero is the dot product between the vectors five, six, nine and 𝐫 is equal to 28.