Video Transcript
Find the vector equation of the
straight line whose slope is negative eight-thirds and passes through the point
four, negative nine. Is it (A) 𝐫 is equal to negative
nine, four plus 𝑘 multiplied by three, negative eight? (B) 𝐫 is equal to four, negative
nine plus 𝑘 multiplied by eight, negative three. (C) 𝐫 is equal to four, negative
nine plus 𝑘 multiplied by three, negative eight. Or (D) 𝐫 is equal to three,
negative eight plus 𝑘 multiplied by four, negative nine.
We recall that the vector equation
of a line is 𝐫 is equal to 𝐫 sub zero plus 𝑘 multiplied by 𝐝, where 𝐫 sub zero
is the position vector of any point on the line, 𝐝 is the direction vector of the
line, and 𝑘 is any scalar value.
We are told that the line in this
question passes through the point four, negative nine, which has the position vector
four, negative nine. And in the vector equation of the
line, this is our point with position vector 𝐫 sub zero. Therefore, to find the vector
equation of this line, we only need to find its direction vector 𝐝. We can do this by recalling what is
meant by the slope of a line.
The slope of a line is the change
in 𝑦 divided by the change in 𝑥. Therefore, if the slope of the line
is negative eight-thirds, this means that for every three units we move
horizontally, we must move eight units vertically.
There are two equivalent ways of
writing this as a vector. We can think of moving three units
right and eight units down, giving us the vector three, negative eight, or moving
three units left and eight units up, giving us the vector negative three, eight. In fact, these are equivalent. They are both direction vectors of
the line. Only the vector three, negative
eight appears in the options. So, we will choose this as our
direction vector.
Hence, the vector equation of the
line is 𝐫 is equal to four, negative nine plus 𝑘 multiplied by three, negative
eight. The correct answer is option
(C).
