Video Transcript
By using determinants,
determine which of the following sets of points are collinear. Option (A) 𝐴 negative six,
four; 𝐵 negative eight, four; 𝐶 three, 10. Option (B) 𝐴 negative 10,
negative four; 𝐵 negative eight, negative two; 𝐶 negative five, one. Option (C) 𝐴 negative three,
six; 𝐵 eight, negative seven; 𝐶 negative three, negative eight. Or is it option (D) 𝐴 negative
10, negative six; 𝐵 negative two, one; 𝐶 zero, negative nine?
We first recall that we can
check the collinearity of a triplet of points in the coordinate plane by using
determinants. In particular, if the
determinant of this matrix is zero, then the three points will be collinear. And if the three points are
collinear, then the determinant of this matrix will be zero.
Therefore, we can determine the
collinearity of each triplet by evaluating the determinant of each matrix
generated from the three points. Substituting the coordinates
given in option (A) into this matrix gives us the determinant of the following
three-by-three matrix. We can evaluate the determinant
of this matrix by expanding over the first row and evaluating to obtain negative
12. This is nonzero, so the three
points are not collinear.
We can follow the same process
for the triplet in option (B). We substitute their coordinates
into the matrix, expand on the first row, and evaluate to get zero. Since the determinant of this
matrix is zero, we can conclude that the points must be collinear.
For due diligence, we can also
check the final two options in the same way. For the triplet in option (C),
we find that the determinant of this matrix is negative 154. So the three points are not
collinear. Similarly, in option (D), we
find that the determinant of the generated matrix is negative 94. Hence, only the points in
option (B) are collinear.
