Question Video: Finding the Parametric Equation of the Line with a Given Point and a Given Direction Vector Mathematics

Give the parametric equation of the line on point (2, −4, 4), with direction vector 〈1, −1, 5〉.

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Video Transcript

Give the parametric equation of the line on point two, negative four, four with direction vector one, negative one, five.

We begin by recalling that the parametric equations of a line are as follows. 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 is a point that lies on the line. And the vector 𝐥, 𝐦, 𝐧 is a direction vector of the line. We also have the parameter 𝑡, which is a scalar quantity. In this question, we are told that the point two, negative four, four lies on the line. So these will be the values of 𝑥 sub zero, 𝑦 sub zero, and 𝑧 sub zero, respectively. We are also given a direction vector one, negative one, five, which will be the values of 𝐥, 𝐦, and 𝐧.

Substituting the values of 𝑥 sub zero and 𝐥, we have 𝑥 is equal to two plus 𝑡. Next, we have 𝑦 is equal to negative four minus 𝑡. Finally, substituting the values four and five for 𝑧 sub zero and 𝐧, we have 𝑧 is equal to four plus five 𝑡. The parametric equation of the line on point two, negative four, four with direction vector one, negative one, five is as shown.

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