Question Video: Finding the Parametric Form of the Equation of a Plane That Passes through a Point and Two Vectors Mathematics

Video Transcript

Find, in parametric form, the equation of the plane that passes through the point 𝐴 one, two, one and the two vectors 𝐝 one equals one, negative one, two and 𝐝 two equals two, negative one, one.

Okay, we have here all these possible answers for the parametric form of the equation of our plane. And as we’ve seen, this plane passes through this point 𝐴 and contains these two vectors 𝐝 one and 𝐝 two. Visually then, our plane might look like this. In writing the equation of this plane, the idea is that we start at our known point and then we move out from that point in multiples of the vectors 𝐝 one and 𝐝 two. We could say then that a vector describing our plane’s entire surface — we’ll call it 𝐫 — is equal to a vector to our known point on the plane plus one parameter that varies across all possible numbers multiplied by our one vector 𝐝 one added to another parameter that also varies across all possible numbers multiplied by the vector 𝐝 two.

Even though it may seem we’re looking at just one equation here, actually, there are three involved: one for the 𝑥-dimension, one for the 𝑦, and one for the 𝑧. To see that, we can replace this vector 𝐫 with its components. And now we see that, for example, 𝑥 is equal to one plus 𝑡 one times one plus 𝑡 two times two. Then, likewise, 𝑦 is equal to two plus 𝑡 one times negative one plus 𝑡 two times negative one. And then, lastly, we also have an equation for the 𝑧-component of our vector. 𝑧 equals one plus two times 𝑡 one plus 𝑡 two. This is the parametric form of the equation of our plane. And if we look through our answer options, we see it matches up with option (A). The equation of our plane in parametric form is 𝑥 equals one plus 𝑡 one plus two times 𝑡 two, 𝑦 equals two minus 𝑡 one minus 𝑡 two, 𝑧 equals one plus two 𝑡 one plus 𝑡 two.