Video Transcript
Using the vector form of the
equation of a straight line, identify whether the points negative seven, five;
negative one, two; and five, negative one are collinear.
We recall that a set of points are
said to be collinear if all of the points lie on the same straight line. There are several ways of checking
this. One way is to begin by finding the
equation between a pair of points and then checking whether the third point
satisfies this. We will begin by finding the vector
equation of the line which passes through negative seven, five and negative one,
two. We recall that this line has
equation 𝐫 is equal to 𝑥 sub one, 𝑦 sub one plus 𝑘 multiplied by 𝑥 sub two
minus 𝑥 sub one, 𝑦 sub two minus 𝑦 sub one.
Substituting in the given values,
the right-hand side becomes negative seven, five plus 𝑘 multiplied by negative one
minus negative seven, two minus five. This, in turn, simplifies to
negative seven, five plus 𝑘 multiplied by six, negative three. Grouping the corresponding
components on the right-hand side, we have 𝐫 is equal to negative seven plus six
𝑘, five minus three 𝑘. If the three points are collinear,
our third point five, negative one will lie on this line. We can check this by substituting
its position vector five, negative one into the vector equation of the line. Equating the components, we have
five is equal to negative seven plus six 𝑘 and negative one is equal to five minus
three 𝑘.
We can solve the first equation by
adding seven to both sides and then dividing through by six. This gives us 𝑘 is equal to
two. The second equation also gives us a
solution of 𝑘 equals two. As this value of 𝑘 is the same, we
can therefore conclude that the point five, negative one lies on the line. The correct answer is therefore
yes, the three points are collinear. This can also be shown graphically
on the 𝑥𝑦-plane.
