Question Video: Finding the Parametric Equation of a Line Passing through Two Points on a Cube Mathematics • Third Year of Secondary School

Video Transcript

The given figure is a cube of volume 27 cubic units. Find in parametric form the equation of line 𝐺𝐵. Is it (A) 𝑥 equals three 𝑡, 𝑦 equals negative three 𝑡, and 𝑧 equals three plus three 𝑡? (B) 𝑥 equals three 𝑡, 𝑦 equals three 𝑡, and 𝑧 equals three plus three 𝑡. Option (C) 𝑥 equals three 𝑡, 𝑦 equals three 𝑡, 𝑧 equals three minus three 𝑡. Option (D) 𝑥 equals three minus three 𝑡, 𝑦 equals negative three 𝑡, and 𝑧 equals three 𝑡. Or option (E) 𝑥 equals three minus three 𝑡, 𝑦 is equal to three 𝑡, and 𝑧 equals three 𝑡.

We begin by considering that the diagram shows a cube of volume 27 cubic units. The volume of any cube is equal to the side length cubed. This means that in this question, 𝐿 cubed is equal to 27 cubic units. Cube rooting both sides of this equation gives us 𝐿 equals three. The cube in the figure therefore has side length three units. We’re asked to find the equation of the line 𝐺𝐵 in parametric form. We will begin by finding the coordinates of points 𝐺 and 𝐵.

Point 𝐺 lies on the 𝑧-axis. This means it will have an 𝑥- and 𝑦-coordinate equal to zero. As the cube has side length three, point 𝐺 has coordinates zero, zero, three. Point 𝐵 lies three units in the 𝑥-direction. It also lies three units in the 𝑦-direction. As the point lies in the 𝑥𝑦-plane, the 𝑧-coordinate will equal zero. 𝐵 has coordinates three, three, zero.

Next, we recall that the parametric equations of a line are a nonunique set of three equations of the form 𝑥 is equal to 𝑥 sub zero plus 𝑡𝑙, 𝑦 is equal to 𝑦 sub zero plus 𝑡𝑚, and 𝑧 is equal to 𝑧 sub zero plus 𝑡𝑛, where 𝑥 sub zero, 𝑦 sub zero, 𝑧 sub zero are the coordinates of a point that lies on the line. The vector 𝐥, 𝐦, 𝐧 is a direction vector of the line. And 𝑡 is a real number known as the parameter that varies from negative ∞ to ∞. We can choose either of our points 𝐵 or 𝐺 to be 𝑥 sub zero, 𝑦 sub zero, 𝑧 sub zero. Whilst they would both give us a valid solution, we need to match one of the solutions given.

We will therefore begin by letting point 𝐺 have coordinates 𝑥 sub zero, 𝑦 sub zero, 𝑧 sub zero. Next, we need to calculate a direction vector of the line 𝐺𝐵. One way of doing this is to subtract the position vectors of the two points. We can do this in either order. However, we will subtract the vector zero, zero, three from the vector three, three, zero. The vector three, three, negative three is therefore a direction vector of line 𝐺𝐵.

We will let the three components be 𝐥, 𝐦, and 𝐧, respectively. Substituting in the values of 𝑥 sub zero and 𝐥 into the general form gives us 𝑥 is equal to zero plus three 𝑡. This simplifies to three 𝑡. 𝑦 is also equal to three 𝑡. Substituting our values of 𝑧 sub zero and 𝐧 gives us 𝑧 is equal to three minus three 𝑡. This set of equations matches option (C). As already mentioned, we could have used point 𝐵 or any other point that lies on 𝐺𝐵. We could also have used a different direction vector, for example, negative three, negative three, three. However, from the options in this question, the set of parametric equations of line 𝐺𝐵 are 𝑥 equals three 𝑡, 𝑦 equals three 𝑡, and 𝑧 equals three minus three 𝑡.