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

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

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The figure shows a cube of side length 6. The point π is the midpoint of line segment π΄π΅. Which of the following are parametric equations of the line ππ? [A] π₯ = 6π‘, π¦ = 3π‘, π§ = 3π‘ [B] π₯ = 6π‘, π¦ = 6 β 6π‘, π§ = 3 β 3π‘ [C] π₯ = 6π‘, π¦ = 6π‘, π§ = 3π‘ [D] π₯ = 3π‘, π¦ = 3π‘, π§ = 6π‘ [E] π₯ = 6 β 6π‘, π¦ = 6 β 6π‘, π§ = 3π‘

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### Video Transcript

The figure shows a cube of side length six. The point π is the midpoint of line segment π΄π΅. Which of the following are the parametric equations of the line ππ? Is it (A) π₯ is equal to six π‘, π¦ is equal to three π‘, and π§ is equal to three π‘. (B) π₯ is equal to six π‘, π¦ is equal to six minus six π‘, and π§ is equal to three minus three π‘. Option (C) π₯ is equal to six π‘, π¦ is equal to six π‘, and π§ is equal to three π‘. (D) π₯ is equal to three π‘, π¦ is equal to three π‘, and π§ is equal to six π‘. Or (E) π₯ is equal to six minus six π‘, π¦ is equal to six minus six π‘, and π§ is equal to three π‘.

We are told in the question that we have a cube of side length six. We are told that π is the midpoint of the line segment π΄π΅. And weβre asked to find the parametric equations of the line ππ.

We recall 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 the point with coordinates π₯ sub zero, π¦ sub zero, π§ sub zero lies on the line. 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 know that the origin has coordinates zero, zero, zero. And since this point lies on line ππ, we can let π₯ sub zero, π¦ sub zero, and π§ sub zero all equal zero. In order to work out a direction vector of this line, we firstly need to work out the coordinates of point π. Point π΄ has coordinates six, six, six. And point π΅ has coordinates six, six, zero.

Since π is the midpoint of π΄π΅, we can calculate its coordinates by finding average of the corresponding coordinates of π΄ and π΅. The average of six and six is six, and the average of six and zero is three. Therefore, π has coordinates six, six, three. We are now in a position where we can find a direction vector of the line ππ.

One way of doing this is to subtract the position vectors of the two points. We can do this in either order. In this question, weβll subtract the vector zero, zero, zero from the vector six, six, three. One direction vector of line ππ is therefore equal to six, six, three. These are the values of π, π, and π that we will substitute into the general form.

We could use either point π or point π for π₯ sub zero, π¦ sub zero, π§ sub zero, as both of these lie on the line. As we are looking for a specific solution that matches one of our options, we will use the origin. Substituting our values of π₯ sub zero and π into the general form gives us π₯ is equal to zero plus six π‘. This simplifies to just six π‘. Likewise, we have π¦ is equal to zero plus six π‘ and π§ is equal to zero plus three π‘.

Simplifying our equations, we notice that they match those in option (C). From the options given, the parametric equations of line ππ are π₯ equals six π‘, π¦ equals six π‘, and π§ equals three π‘. As already mentioned, there are many other sets of parametric equations we could have chosen based on the information in this question.

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