Question Video: Finding the Length of the Segment of the 𝑥-Axis Cut off by a Given Plane Mathematics

Video Transcript

What is the length of the segment of the 𝑥-axis cut off by the plane six 𝑥 plus three 𝑦 plus five 𝑧 equals four?

In this example, we have a plane, and we’re told that this plane intersects the 𝑥-axis of our coordinate frame. Say that intersection happens here. When our question asks what the length of the segment of the 𝑥-axis cut off by the plane is, it’s asking what the distances from the origin of our frame to this point of intersection. So the question is, where does this plane intersect the 𝑥-axis? Since we’re talking about intercepts between planes and axes, we can recall the intercept form of the equation of a plane. Written this way, the values 𝐴, 𝐵, and 𝐶 correspond to the 𝑥-, 𝑦-, and 𝑧-values of the points of intersection along those respective axes.

For example, we would say that the coordinates of the point of intersection between the plane and the 𝑥-axis are 𝐴: zero, zero. If we can solve for 𝐴, then we’ll have the answer to our question. Our task then will be to rearrange the given equation of our plane so that it’s in intercept form. Once it’s written that way, we can identify the value of 𝐴. Notice that the intercept form of a plane’s equation has the value of one by itself on one side. We can create a similar situation in our given plane equation by dividing both sides by four. If we do this, we get three-halves 𝑥 plus three-quarters 𝑦 plus five-fourths 𝑧 equaling one.

And now, to force this form of our equation into intercept form, we’ll write it in a bit of a strange way. We write it as 𝑥 divided by two-thirds plus 𝑦 divided by four-thirds plus 𝑧 over four-fifths equaling one. Mathematically, these two ways of writing our plane’s equation are the same. We use this second way, though, to clarify what values are equal to 𝐴, 𝐵, and 𝐶 in our intercept form. 𝐶 is equal to four-fifths, 𝐵 is equal to four-thirds, and 𝐴, the value we want to solve for, is two-thirds. We say then the length of the segment of the 𝑥-axis cut off by this plane is two-thirds.