Video Transcript
Which of the following sets of parametric equations describes the straight line that passes through the point 𝐴 with coordinates two, three, four and the origin? Is it option (A) 𝑥 is equal to two minus two 𝑡, 𝑦 is equal to three plus three 𝑡, and 𝑧 is equal to negative four minus four 𝑡? Option (B) 𝑥 is equal to two minus two 𝑡, 𝑦 is equal to three minus three 𝑡, and 𝑧 is equal to four minus four 𝑡. Option (C) 𝑥 is equal to negative two plus 𝑡, 𝑦 is equal to negative three plus 𝑡, and 𝑧 is equal to four plus 𝑡. Option (D) 𝑥 is equal to two 𝑡, 𝑦 is equal to 𝑡, and 𝑧 is equal to negative three 𝑡. Or option (E) 𝑥 is equal to 𝑡, 𝑦 is equal to negative three 𝑡, and 𝑧 is equal to 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 positive ∞. In this question, we are told that our line passes through the point two, three, four and the origin. We know that the origin has coordinates zero, zero, zero. We could therefore use either of these points for 𝑥 sub zero, 𝑦 sub zero, 𝑧 sub zero.
We will begin by selecting the point with coordinates two, three, four. However, if this answer does not match one of the options, we can try the point zero, zero, zero. As we know two points that lie on the line, we can calculate the direction vector by subtracting the position vectors of these two points. We can do this in either order. We could subtract the vector zero, zero, zero from two, three, four. Or we can subtract the vector two, three, four from zero, zero, zero. Using the second method gives us the direction vector negative two, negative three, negative four. We now have values of 𝑥 sub zero, 𝑦 sub zero, 𝑧 sub zero together with 𝑙, 𝑚, and 𝑛.
Substituting them into the general form, we have 𝑥 is equal to two minus two 𝑡, 𝑦 is equal to three minus three 𝑡, and 𝑧 is equal to four minus four 𝑡. From the five options given, this matches option (B). The set of parametric equations that describes the straight line that passes through the point two, three, four and the origin is 𝑥 equals two minus two 𝑡, 𝑦 is equal to three minus three 𝑡, and 𝑧 is equal to four minus four 𝑡.
As already mentioned, we could use the origin as the point that lies on the line together with the direction vector two, three, four. However, using these does not match any of the options given. Whilst the parametric equations are not unique, in this question, we had to select a specific set.
