Magnitudes, Extremes, Bisectors and Geometry

Transcript

0:00Introduction

Hello everyone! Let us start with a small puzzle. Take a parallelogram with sides five and four, and one diagonal of length seven. How long is the other diagonal? No angle is given. Looks impossible, right? Today you will find it in one line, with the dot product. We will also find a greatest value without calculus, and the direction that cuts an angle in half. Questions like these do come up in JEE papers. So, let's begin.

0:29The parallelogram law

So, we start from what you already know. The small arrows on top mark a vector and b vector, and the two vertical bars give the modulus, the length of the vector, read as mod a, okay? The modulus of a plus b, squared, equals the modulus of a squared, plus the modulus of b squared, plus two times a dot b. For a minus b, only the last sign flips: minus two times a dot b. Right? Now, what happens if we simply add these two lines? Yes, exactly, the two a dot b terms cancel. So, the modulus of a plus b squared, plus the modulus of a minus b squared, equals two times, the modulus of a squared plus the modulus of b squared. This is the parallelogram law, and there is no angle anywhere in it! Clear? And here is why it is called the parallelogram law. In the parallelogram with sides a vector and b vector, a plus b and a minus b are exactly the two diagonals. So the squares of the diagonals add up to twice the sum of the squares of the sides. Know any three lengths, and the fourth is yours. Got it?

1:39Cauchy–Schwarz: how big can a dot product get?

Now the second big idea. The dot product, a dot b, equals the modulus of a, times the modulus of b, times cos theta, where theta is the angle between them. And cos theta lies between minus one and one; the sign with a small line under it means less than or equal to. Okay? So, with the lengths fixed, how big can a dot b get? Very good, at most the modulus of a times the modulus of b, when cos theta is one. So the modulus of a dot b is less than or equal to the modulus of a times the modulus of b. This is the Cauchy Schwarz inequality. Equality holds if and only if, written i f f, the two vectors are parallel, the sign with two short vertical lines. With me? Now it becomes a weapon. Let a vector have the components p, q, r, in round brackets, and b vector the components x, y, z. Then a dot b is p times x, plus q times y, plus r times z. Right? So, for a fixed x squared plus y squared plus z squared, the max, the greatest value, of this expression is the root of p squared plus q squared plus r squared, times the root of x squared plus y squared plus z squared. It is reached when b vector points the same way as a vector. And we used no calculus at all. Clear?

3:09Two unit vectors: sum, difference and bisectors

Next, two unit vectors, of length one, at an angle theta. The modulus of a plus b, squared, is one plus one plus two cos theta, that is, two plus two cos theta, which is four times cos squared of, theta over two. Okay? Take the root, and the modulus of a plus b is two times cos of, theta over two. And with a minus sign, the modulus of a minus b is two times sine of, theta over two. Got it? Now, the angle bisector. For any two vectors, not necessarily of the same length, does a plus b always point along the bisector of the angle between them? No, no, careful, that is the trap! Only when the two lengths are equal. If one vector is much longer, the sum leans towards it. So first we make both lengths one. The unit vector a hat, with the little cap, is a vector over the modulus of a, and the same for b hat, for non zero vectors. Right? Now the parallelogram on a hat and b hat is a rhombus, and the diagonal of a rhombus cuts the angle in half. So the internal bisector is along a hat plus b hat, and the external bisector along a hat minus b hat. Clear? And these two bisectors are always perpendicular. Dot them: the bracket, a hat plus b hat, dotted with the bracket, a hat minus b hat, is one minus one, which is zero. For non zero vectors, a zero dot product means perpendicular. With me?

4:37One more fact: equal, perpendicular vectors

One last quick fact. Take three vectors, a, b and c, that are mutually perpendicular, and all of the same length. Okay? Then each one makes the same angle with their sum, a plus b plus c: cos inverse of, one over root three. The small minus one on the cos means cos inverse. Right?

4:59Example 1: find mod a minus b with the parallelogram law

Now, example one, our puzzle from the start. The modulus of a is five, the modulus of b is four, and the modulus of a plus b is seven. Find the modulus of a minus b, the other diagonal. Pause the video and try it first. Done? Okay, let's check. Some students write five minus four, which is one. Is that right? No, no, careful, the modulus of a difference is not the difference of the moduli. Use the parallelogram law: the modulus of a minus b, squared, equals two times, twenty five plus sixteen, minus forty nine. Those are the squares of five, four and seven. Okay? Two times forty one is eighty two. And eighty two minus forty nine gives thirty three. Right? So the modulus of a minus b is root thirty three. Puzzle solved, and we never needed the angle! See how fast that was?

5:59Example 2: greatest value of 2x+3y-6z if x2+y2+z2 = 4

Now a harder one, JEE level. Find the greatest value of two times x, plus three times y, minus six times z, given that x squared plus y squared plus z squared equals four. Pause the video and try it first. Done? Alright, let's solve it. The expression is a dot product: the vector two, three, minus six, dotted with the vector x, y, z. So by Cauchy Schwarz, it is at most the product of the two lengths. With me? So, what is the length of the vector two, three, minus six? It is the root of four plus nine plus thirty six, which is root forty nine, so seven. The minus sign goes away when we square. Okay? Now the vector x, y, z. We are given that x squared plus y squared plus z squared is four. So its length is four, right? Oh no, careful, four is the square of the length, so the length is root four, which is two. If you use four here, you get twenty eight, and that is wrong. Got it? So the expression is at most seven times two, which is fourteen. Clear? And fourteen is actually reached. It happens when x, y, z points the same way as two, three, minus six, that is, x, y, z equals two sevenths of, two, three, minus six. So the greatest value is fourteen, without any calculus!

7:34Quick revision: parallelogram law, Cauchy-Schwarz, bisectors

So, quick revision. The parallelogram law: the modulus of a plus b squared, plus the modulus of a minus b squared, is two times the sum of the squares of the moduli. Cauchy Schwarz: the modulus of a dot b is at most the modulus of a times the modulus of b, with equality only for parallel vectors. And the bisectors are along a hat plus b hat, internal, and a hat minus b hat, external. Now go and try the practice problems on this page. All the best!