Video Transcript
Using determinants, are the points zero, one; two, one-half; and four, zero collinear?
In this question, we need to determine whether three given points are collinear. That means, do they lie on the same straight line? And there’s a few different ways we could go about doing this. For example, we could find an equation of a straight line between a pair of these two points and then determine whether the third point lies on this line. However, the question wants us to do this by using determinants. And to do this, we need to recall the following fact about determinants. If we have three distinct points 𝑥 sub one, 𝑦 sub one; 𝑥 sub two, 𝑦 sub two; and 𝑥 sub three, 𝑦 sub three, then we can determine whether these points are collinear by calculating the determinant of the three-by-three matrix 𝑥 sub one, 𝑦 sub one, one, 𝑥 sub two, 𝑦 sub two, one, 𝑥 sub three, 𝑦 sub three, one.
If this determinant is zero, then the three points are collinear. If this determinant is nonzero, then the three points are noncollinear. And it is worth noting the statement does work in both directions. If the determinant is zero, then the points are collinear. And similarly, if the points are collinear, then the determinant is zero, where we assume we have three distinct points. And to see why this is true, we can recall the absolute value of this determinant gives us the area of a parallelogram with these three points as vertices. And the only way the area of a parallelogram can be zero is if its vertices are collinear. Therefore, we can determine whether these three points are collinear by substituting the three points given to us in the question into this equation.
We need to determine whether the determinant of the matrix zero, one, one, two, one-half, one, four, zero, one is equal to zero. And we can evaluate the determinant of this matrix in any way we choose. We’re going to expand over the first row because it includes the number zero. To expand over the first row in this matrix, we need to find all of the matrix minors of this row. And remember, the sign of this expansion will change depending on the parity. In this case, the second term will be negative. This gives us zero times the determinant of the matrix one-half, one, zero, one minus one times the determinant of the matrix two, one, four, one plus one times the determinant of the matrix two, one-half, four, zero.
Now, all that’s left to do is evaluate this expression. The first term has a factor of zero, so it’s equal to zero. And remember, to evaluate the determinant of a two-by-two matrix, we need to find the difference in the products of the diagonals. In the second term, that’s two times one minus four times one, which is two minus four. And in the third term, that’s two times zero minus four times one-half, which is zero minus two. This gives us negative one times two minus four plus one times zero minus two. And if we evaluate this expression, we get two minus two, which is equal to zero. And since this determinant is equal to zero, the three points are either not distinct or collinear. And we can see that they are distinct points, so they must be collinear.
Therefore, we were able to show by using the determinants, the points zero, one; two, one-half; and four, zero are collinear.