Video Transcript
Find the vector equation of the
straight line passing through the points six, negative seven and negative four,
six. Is it (A) 𝐫 is equal to six,
negative seven plus 𝑘 multiplied by 10, negative 13? (B) 𝐫 is equal to negative four,
six plus 𝑘 multiplied by negative 13, 10. (C) 𝐫 is equal to six, negative
four plus 𝑘 multiplied by negative seven, six. Or (D) 𝐫 is equal to negative
four, six plus 𝑘 multiplied by 10, 13.
We begin by recalling that the
vector equation of a straight line is written in the form 𝐫 is equal to 𝐫 sub zero
plus 𝑘 multiplied by 𝐝, where 𝐫 sub zero is the position vector of any point that
lies on the line, 𝐝 is the direction vector of the line, and 𝑘 is any scalar. We are given the coordinates of two
points, six, negative seven and negative four, six, that both lie on the line. And we can use either of these as
the position vector to help find the vector equation of the line. In option (A), the position vector
is six, negative seven. And in options (B) and (D), the
position vector is negative four, six.
Let’s begin by letting the position
vector 𝐫 sub zero be six, negative seven. We now need to find the direction
vector given the two points that lie on the line. We recall that the slope of any
line is equal to the change in 𝑦 over the change in 𝑥. Substituting in the points given,
we see that the slope is equal to negative seven minus six over six minus negative
four. This is equal to negative 13 over
10. Recalling that any line with slope
𝑚 equal to 𝑝 over 𝑞 has direction vector 𝑞, 𝑝, we see that the direction vector
here is equal to 10, negative 13. As this is one possible direction
vector of our straight line, we have 𝐫 is equal to six, negative seven plus 𝑘
multiplied by 10 negative 13.
We noticed that this corresponds to
option (A), proving that this is the vector equation of the straight line passing
through the points six, negative seven and negative four, six. Looking at the other options, we
recall that options (B) and (D) do indeed pass through the point negative four,
six. However, they do not have a
direction vector which is equal to or parallel to 10, negative 13. We can therefore rule out options
(B) and (D). The direction vector of option (C)
is negative seven, six. And this too is not parallel to the
direction vector 10, negative 13.
At this point, it is worth
recalling that the vector equation of a straight line is not unique. Other possible vector equations of
the straight line from the information given are 𝐫 is equal to negative four, six
plus 𝑘 multiplied by 10, negative 13; 𝐫 is equal to six, negative seven plus 𝑘
multiplied by negative 10, 13; and 𝐫 is equal to negative four, six plus 𝑘
multiplied by negative 10, 13. The direction vectors in the last
two options move in the opposite direction. Any of these four solutions are
valid. However, the only one that was
given as one of the options is 𝐫 is equal to six, negative seven plus 𝑘 multiplied
by 10, negative 13.
