### Video Transcript

The given figure shows a rectangular prism where the coordinates of πΆ and πΉ are six, zero, seven and zero, five, seven, respectively. Which of the following is the equation of line πΆπΈ in parametric form? Is it (A) π₯ equals six π‘, π¦ equals five minus five π‘, and π§ equals seven minus seven π‘? (B) π₯ equals six minus six π‘, π¦ equals five π‘, and π§ equals seven π‘. Option (C) π₯ equals six π‘, π¦ equals five π‘, and π§ equals seven π‘. (D) π₯ equals six minus six π‘, π¦ equals five minus five π‘, and π§ equals seven π‘. Or option (E) π₯ equals six minus six π‘, π¦ equals five π‘, and π§ equals seven minus seven π‘.

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 πΆ are six, zero, seven. The coordinates of point πΉ are zero, five, seven. In this part of the question, we are asked to find the parametric equation of the line πΆπΈ. 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 π‘π, π¦ is equal to π¦ sub zero plus π‘π, and π§ is equal to π§ sub zero plus π‘π, where π₯ sub zero, π¦ sub zero, π§ sub zero are the coordinates of a point that lies on the line. The vector π, π, π is a direction vector of the line. And π‘ is a real number known as the parameter that varies from negative β to β.

We already know that the point six, zero, seven lies on line πΆπΈ. This means we can let these be the values of π₯ sub zero, π¦ sub zero, and π§ sub zero. In order to find a direction vector of line πΆπΈ, we firstly need to work out the coordinates of point πΈ. Point πΈ lies on the π¦-axis. This means that its π₯- and π§-coordinates will be zero. Point πΈ is the same distance along the π¦-axis as point πΉ. This means it will have a π¦-coordinate of five. The coordinates of point πΈ 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 π, π, and π in the general form.

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

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 πΈ instead of point πΆ for π₯ sub zero, π¦ sub zero, π§ 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.

Letβs 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 π·πΉ in parametric form? Is it option (A) π₯ equals six minus six π‘, π¦ equals five π‘, π§ equals seven π‘? (B) π₯ equals six π‘, π¦ equals five minus five π‘, π§ equals seven π‘. (C) π₯ equals six minus six π‘, π¦ equals five minus five π‘, π§ equals seven minus seven π‘. (D) π₯ equals negative six π‘, π¦ equals five π‘, π§ equals seven π‘. Or option (E) π₯ equals six π‘, π¦ equals five π‘, π§ equals seven minus seven π‘.

We are now trying to find a set of parametric equations for the line π·πΉ. We know that this line passes through the point with coordinates zero, five, seven. Point π· Lies on the π₯-axis and has moved in the π₯-direction the same distance as point πΆ. This means that point π· has coordinates six, zero, zero. We can use the coordinates of either point π· or point πΉ for π₯ sub zero, π¦ sub zero, π§ 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 π, π, and π, respectively.

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