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

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

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Write the parametric equations of a straight line 𝐿 passing through the point 𝑝₁ = (3, 3, −1) and the midpoint between 𝑝₂ = (1, −1, 1) and 𝑝₃ = (3, 5, 5).

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Video Transcript

Write the parametric equations of a straight line 𝐿 passing through the point 𝑝 sub one, which is equal to three, three, negative one, and the midpoint between 𝑝 sub two, which is equal to one, negative one, one, and 𝑝 sub three, which is equal to three, five, five. Is it option (A) 𝑥 is equal to three minus 𝑡, 𝑦 is equal to three minus 𝑡, and 𝑧 is equal to negative one plus four 𝑡? Is it option (B) 𝑥 is equal to three minus two 𝑡, 𝑦 is equal to three minus 𝑡, and 𝑧 is equal to negative one plus four 𝑡? Is it option (C) 𝑥 is equal to three minus 𝑡, 𝑦 is equal to three minus 𝑡, and 𝑧 is equal to negative one plus two 𝑡? Option (D) 𝑥 is equal to negative two plus three 𝑡, 𝑦 is equal to three minus 𝑡, and 𝑧 is equal to negative one plus four 𝑡. Or option (E) 𝑥 is equal to three minus two 𝑡, 𝑦 is equal to three minus 𝑡, and 𝑧 is equal to negative one plus two 𝑡.

We begin by recalling 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. 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 ∞.

In this question, we are given a point that lies on the line. It has coordinates three, three, negative one. We will let this point be 𝑥 sub zero, 𝑦 sub zero, and 𝑧 sub zero. We are also told that the line passes through the midpoint of 𝑝 sub two and 𝑝 sub three. We can find the midpoint of any two points in three dimensions by finding the average of their corresponding coordinates. The 𝑥-coordinates of 𝑝 sub two and 𝑝 sub three are one and three, respectively. This means that the 𝑥-coordinate of the midpoint will be equal to one plus three divided by two.

We can repeat this for the 𝑦- and 𝑧-coordinates as shown. One plus three is equal to four, and dividing this by two gives us two. Negative one plus five is also equal to four, so dividing this by two also gives us two. Finally, one plus five divided by two is equal to three. The coordinates of the midpoint of 𝑝 sub two and 𝑝 sub three are two, two, three.

We now have the coordinates of two points that lie on the line. And we can use these to calculate a direction vector. One way of doing this is by subtracting the vector three, three, negative one from the vector two, two, three. Subtracting the corresponding components gives us the vector negative one, negative one, four.

We will use these values for 𝐥, 𝐦, and 𝐧 in the general form. Firstly, we have 𝑥 is equal to three plus negative one 𝑡, which simplifies to three minus 𝑡. 𝑦 is also equal to three plus negative one 𝑡, which we can once again simplify to three minus 𝑡. 𝑧 is equal to negative one plus four 𝑡. One set of parametric equations of the straight line 𝐿, which passes through point 𝑝 sub one and the midpoint between 𝑝 sub two and 𝑝 sub three, is 𝑥 equals three minus 𝑡, 𝑦 is equal to three minus 𝑡, and 𝑧 is equal to negative one plus four 𝑡. This corresponds to option (A) in the question.

It is worth noting a couple of other solutions we could’ve found from the information in the question. Firstly, we could’ve used the midpoint with coordinates two, two, three as 𝑥 sub zero, 𝑦 sub zero, and 𝑧 sub zero. We could also have subtracted the vectors in the other order when finding the direction vector. This would’ve given us a direction vector of one, one, negative four. Using a combination of any of these would’ve given a valid solution. However, none of these match the options given in this question.

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