### 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.