Area, Collinearity and Calculus with Determinants

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle before we start. Can a triangle have zero area? Strange question, right? Hold on to it; we will answer it in a few minutes. Today, one determinant does three jobs: it gives the area of a triangle, it tells us when three points lie on one line, and it gives the equation of a line. And at the end, we will even differentiate a determinant! Questions like these come up often in JEE, so let's begin.

0:33Area of a triangle from its corners

So, look at the figure: the usual x and y axes, and a triangle with corners A, B and C, each marked with a dot. Corner A is the point x one, y one, written in round brackets; the small one below is a subscript, it just means point number one. Then B is x two, y two, and C is x three, y three. Six numbers, and from them we want the area. Here is the formula, also shown in the figure. Delta, the Greek capital letter that looks like a little triangle, stands for the area. Delta equals one half, times a determinant with a modulus around it. See the two pairs of bars: the inner straight bars make the determinant, and the outer bars are the modulus. Each row is one corner: its x, its y, and then a one. Write this down, okay? Now tell me. Why do we need that modulus outside the determinant? Yes, exactly! Because a determinant can be negative, but an area never can. Without the modulus, the determinant is positive when you go from A to B to C anticlockwise; the arrows on screen show that order, and the grey text in the figure says the same. Clockwise is just a swap of two rows, and a swap changes the sign, remember? So always take the modulus, and never forget the half. Clear?

1:55Zero area means collinear

Now, back to our puzzle. Imagine corner C sliding down, slowly, until A, B and C all sit on one straight line. What happens to the area of the triangle? Zero! Correct! The triangle is squashed flat. So, three points are collinear exactly when this determinant is zero. The double arrow means if and only if: it works both ways. The orange text in the figure says exactly this. So yes, a triangle can have zero area, but then it is not really a triangle, it is a line! Got it? Now, a lovely trick from this. Take two fixed points, x one, y one, and x two, y two. Any point x, y on the line through them is collinear with both of them, right? So what equation must x and y satisfy? Yes, exactly! The determinant must be zero. First row x, y, one; then the two given points, each with a one; set it equal to zero. Expand it, and the equation of the line comes out directly. No slope needed. With me?

3:03Calculus with determinants

Alright, now something new. What if the entries are not fixed numbers, but functions of x? Then the determinant is also a function of x, right? So how would you differentiate it? Here is the rule. Write the determinant by its rows: capital R one, R two, R three. The small dash at the top right is a prime: R one prime means row one, differentiated. The d over d x in front means the derivative. It equals three determinants added: in the first, only row one is differentiated; in the second, only row two; in the third, only row three. Just like the product rule, one factor at a time. Got it? Now, a shortcut question. Can I save time, and differentiate all three rows together, in one single determinant? No, no, careful! That is the trap. One row at a time: three separate determinants, then add them up. And if columns are easier, one column at a time gives the same answer. Okay? And integration? That works in a special case: when only one row depends on x. Then integrate just that row, and keep the other rows as they are. We call it integrating row-wise. Clear?

4:17Example 1: area of the triangle (2,7), (1,1), (10,8)

Now, Example one, an easy one to warm up. Find the area of the triangle with vertices two, seven; one, one; and ten, eight. Pause the video, and try it first. Done? Okay, let's check. Each vertex becomes one row, with a one at the end: two, seven, one; then one, one, one; then ten, eight, one. Expand along row one, with the chessboard signs, plus, minus, plus. Two times, in brackets, one minus eight; minus seven times, one minus ten; and plus, in brackets, eight minus ten. Right? Now, slowly here. Minus seven, times one minus ten. What do you get? Plus sixty three! Very good. One minus ten is minus nine, and minus seven times minus nine is plus sixty three. And two times minus seven is minus fourteen. So, minus fourteen, plus sixty three, minus two: forty seven. Positive, so these vertices are listed anticlockwise. Okay? So the area is forty seven, right? No, no, wait! Where is the half? The area is one half, times the modulus of forty seven: forty seven over two. Forgetting the half is a very common slip. Don't do that!

5:37Example 2: find f(x) and f'(x) for a determinant

Now a harder one. f of x is this determinant. Row one: x, x squared, x cubed; the small two and three up top are powers. Row two: one, two x, three x squared. Row three: zero, two, six x. Find f of x, and f prime of x, its derivative; the dash is the prime. Pause the video and try it. Done? Okay, let's check. Look at row one: x, x squared, x cubed. What is common to all three entries? x, of course! Take x out of row one, just once, and row one becomes one, x, x squared. Now make a zero: R two becomes R two minus R one; the arrow means becomes. One minus one is zero, two x minus x is x, and three x squared minus x squared is two x squared. So row two is now zero, x, two x squared. With me? Now column one is one, zero, zero; capital C one means column one. Expand down it, and only the one survives, times its minor: x times six x, minus two x squared times two. So f of x equals x, times six x squared minus four x squared: two x cubed. Simple! Now f prime, by our row rule: three determinants. But first, look closely at the original determinant. Differentiate its row one: x, x squared, x cubed. What do you get? One, two x, three x squared: that is exactly row two! So R one prime equals R two, and the first determinant has two equal rows: zero. And R two prime is zero, two, six x, exactly row three. Zero again! Each row is the derivative of the row above. Beautiful, right? So only the third one is left. Row three differentiated is zero, zero, six. Expand along that row: six, times x times two x, minus x squared times one. So f prime of x is zero plus zero plus six times x squared: six x squared. And that is exactly the derivative of two x cubed. Both ways agree. Well done!

7:53Quick revision: area, collinearity, derivative

So, quick revision. Area of a triangle: half the modulus of the determinant whose rows are x, y, one, for each corner. Determinant zero means the points are collinear; and for a line, put x, y, one as row one, and set it to zero. To differentiate a determinant, go one row at a time, then add. Now go and try the practice problems on this page. You will enjoy them. All the best!