Question Video: Finding the Parametric Equation of a Line Passing through the Origin with a Given Direction Vector Mathematics

Give the parametric equation of the line through the origin with direction vector <5, −1, 4>.

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Video Transcript

Give the parametric equation of the line through the origin with direction vector five, negative one, four.

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. The vector 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 the line passes through the origin, so it passes through the point with coordinates zero, zero, zero. The line has a direction vector five, negative one, four. We can now substitute these values into the general form of the parametric equations. 𝑥 is equal to zero plus five 𝑡, which simplifies to five 𝑡. 𝑦 is equal to zero plus negative one 𝑡. This simplifies to negative 𝑡. And finally, 𝑧 is equal to zero plus four 𝑡, which is equal to four 𝑡.

The parametric equation of the line passing through the origin with direction vector five, negative one, four is 𝑥 equals five 𝑡, 𝑦 equals negative 𝑡, and 𝑧 equals four 𝑡.