Question Video: Finding the Parametric Equation of a Line Passing through Two Points on a Cube Mathematics

The given figure shows a rectangular prism, where the coordinates of 𝐢 and 𝐹 are (6, 0, 7) and (0, 5, 7) respectively. Which of the following is the equation of line 𝐢𝐸 in parametric form? [A] π‘₯ = 6𝑑, 𝑦 = 5 βˆ’ 5𝑑, 𝑧 = 7 βˆ’ 7𝑑 [B] π‘₯ = 6 βˆ’ 6𝑑, 𝑦 = 5𝑑, 𝑧 = 7𝑑 [C] π‘₯ = 6𝑑, 𝑦 = 5𝑑, 𝑧 = 7𝑑 [D] π‘₯ = 6 βˆ’ 6𝑑, 𝑦 = 5 βˆ’ 5𝑑, 𝑧 = 7𝑑 [E] π‘₯ = 6 βˆ’ 6𝑑, 𝑦 = 5𝑑, 𝑧 = 7 βˆ’ 7𝑑 Which of the following is the equation of line 𝐷𝐹 in parametric form? [A] π‘₯ = 6 βˆ’ 6𝑑, 𝑦 = 5𝑑, 𝑧 = 7𝑑 [B] π‘₯ = 6𝑑, 𝑦 = 5 βˆ’ 5𝑑, 𝑧 = 7𝑑 [C] π‘₯ = 6 βˆ’ 6𝑑, 𝑦 = 5 βˆ’ 5𝑑, 𝑧 = 7 βˆ’ 7𝑑 [D] π‘₯ = βˆ’6𝑑, 𝑦 = 5𝑑, 𝑧 = 7𝑑 [E] π‘₯ = 6𝑑, 𝑦 = 5𝑑, 𝑧 = 7 βˆ’ 7𝑑.

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Video Transcript

The given figure shows a rectangular prism where the coordinates of 𝐢 and 𝐹 are six, zero, seven and zero, five, seven, respectively. Which of the following is the equation of line 𝐢𝐸 in parametric form? Is it (A) π‘₯ equals six 𝑑, 𝑦 equals five minus five 𝑑, and 𝑧 equals seven minus seven 𝑑? (B) π‘₯ equals six minus six 𝑑, 𝑦 equals five 𝑑, and 𝑧 equals seven 𝑑. Option (C) π‘₯ equals six 𝑑, 𝑦 equals five 𝑑, and 𝑧 equals seven 𝑑. (D) π‘₯ equals six minus six 𝑑, 𝑦 equals five minus five 𝑑, and 𝑧 equals seven 𝑑. Or option (E) π‘₯ equals six minus six 𝑑, 𝑦 equals five 𝑑, and 𝑧 equals seven minus seven 𝑑.

There is also a second part to this question which we will look at later. We are told in the question that the coordinates of point 𝐢 are six, zero, seven. The coordinates of point 𝐹 are zero, five, seven. In this part of the question, we are asked to find the parametric equation of the line 𝐢𝐸. We begin by recalling that the parametric equations of a line are a nonunique set of three equations of the form π‘₯ is equal to π‘₯ sub zero plus 𝑑𝑙, 𝑦 is equal to 𝑦 sub zero plus π‘‘π‘š, and 𝑧 is equal to 𝑧 sub zero plus 𝑑𝑛, where π‘₯ sub zero, 𝑦 sub zero, 𝑧 sub zero are the coordinates of a point that lies on the line. The vector 𝑙, π‘š, 𝑛 is a direction vector of the line. And 𝑑 is a real number known as the parameter that varies from negative ∞ to ∞.

We already know that the point six, zero, seven lies on line 𝐢𝐸. This means we can let these be the values of π‘₯ sub zero, 𝑦 sub zero, and 𝑧 sub zero. In order to find a direction vector of line 𝐢𝐸, we firstly need to work out the coordinates of point 𝐸. Point 𝐸 lies on the 𝑦-axis. This means that its π‘₯- and 𝑧-coordinates will be zero. Point 𝐸 is the same distance along the 𝑦-axis as point 𝐹. This means it will have a 𝑦-coordinate of five. The coordinates of point 𝐸 are zero, five, zero.

We can calculate the direction vector of a line if we are given two points that lie on the line. We simply subtract their position vectors. We can do this in either order. In this case, we will subtract the vector six, zero, seven from the vector zero, five, zero. This gives us a direction vector negative six, five, negative seven. We will let these values be 𝑙, π‘š, and 𝑛 in the general form.

Substituting our values of π‘₯ sub zero and 𝑙, we have π‘₯ is equal to six minus six 𝑑. Substituting our values of 𝑦 sub zero and π‘š, we have 𝑦 is equal to zero plus five 𝑑, which simplifies to five 𝑑. Finally, we have 𝑧 is equal to seven minus seven 𝑑. This set of parametric equations matches option (E) from the question. We can therefore conclude that one set of parametric equations of the line 𝐢𝐸 is π‘₯ equals six minus six 𝑑, 𝑦 is equal to five 𝑑, and 𝑧 is equal to seven minus seven 𝑑.

As already mentioned, it is important to note that this is not a unique solution. For example, from the information given, we could have used the point 𝐸 instead of point 𝐢 for π‘₯ sub zero, 𝑦 sub zero, 𝑧 sub zero as this point also lies on the line. We could also have subtracted the position vectors in the opposite order, giving us a direction vector of six, negative five, seven. Using either of these would give us a valid solution; however, they would not match one of the options given.

Let’s now look at the second part of the question. This part of the question is similar to the first, so we have left some of the working on screen.

Which of the following is the equation of line 𝐷𝐹 in parametric form? Is it option (A) π‘₯ equals six minus six 𝑑, 𝑦 equals five 𝑑, 𝑧 equals seven 𝑑? (B) π‘₯ equals six 𝑑, 𝑦 equals five minus five 𝑑, 𝑧 equals seven 𝑑. (C) π‘₯ equals six minus six 𝑑, 𝑦 equals five minus five 𝑑, 𝑧 equals seven minus seven 𝑑. (D) π‘₯ equals negative six 𝑑, 𝑦 equals five 𝑑, 𝑧 equals seven 𝑑. Or option (E) π‘₯ equals six 𝑑, 𝑦 equals five 𝑑, 𝑧 equals seven minus seven 𝑑.

We are now trying to find a set of parametric equations for the line 𝐷𝐹. We know that this line passes through the point with coordinates zero, five, seven. Point 𝐷 Lies on the π‘₯-axis and has moved in the π‘₯-direction the same distance as point 𝐢. This means that point 𝐷 has coordinates six, zero, zero. We can use the coordinates of either point 𝐷 or point 𝐹 for π‘₯ sub zero, 𝑦 sub zero, 𝑧 sub zero. In order to try and match one of the answer options, we will use the point six, zero, zero. However, it is important to note we would get a perfectly valid solution if we use the point zero, five, seven.

To calculate a direction vector of this line, we will subtract the position vectors. Once again, we can do this in either order. In this question, we will subtract the vector six, zero, zero from the vector zero, five, seven. This gives us a direction vector of negative six, five, seven. These are our values of 𝑙, π‘š, and 𝑛, respectively.

The set of parametric equations are therefore as follows. Firstly, π‘₯ is equal to six minus six 𝑑. We have 𝑦 is equal to zero plus five 𝑑, which is just equal to five 𝑑. Finally, we get 𝑧 is equal to seven 𝑑. This set of parametric equations matches option (A). So, we can therefore conclude that one set of parametric equations for the line 𝐷𝐹 are π‘₯ is equal to six minus six 𝑑, 𝑦 is equal to five 𝑑, and 𝑧 is equal to seven 𝑑.

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