The Dot Product

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle first. Go to a corner of your room. From that corner, draw one arrow along the diagonal of a square floor tile, and another along the diagonal of a same-size square tile on the wall. What is the angle between these two arrows? Forty five degrees? Ninety? You cannot even put a protractor there! Today's tool, the dot product, answers it in a few lines. And in physics, work and power are dot products, so this comes up often in JEE papers. So, let's begin.

0:37A dot B: multiply two vectors, get a number

So, look at the left half of the picture. Here, B vector, the letter with a small arrow on top, lies along the bottom, and A vector goes up at an angle theta, this Greek letter. The right half is the cross product, for the next lesson, okay? Now shine a light straight down on A vector, along the dashed line. Its shadow on B vector is marked by the curly bracket, and the label says A cos theta. Why cos theta? Yes, exactly! The dashed line meets B vector at a right angle, so we get a right triangle, and the side next to theta is A cos theta. Plain A, without the arrow, is the length of A vector, and plain B is the length of B vector. Multiply this shadow by B, and you get the dot product, read as A dot B, for the small dot in the middle. So, A dot B equals A B cos theta. Right? And look: the two lengths and cos theta are plain numbers, so the answer has no direction. It is a scalar, which is why this is also called the scalar product.

1:38Dot product from components

Got it? Now, components. Start with the unit vectors: i cap, j cap and k cap, the letters with little caps on top, each of length one. So tell me, what is i cap dot i cap? Correct! The angle is zero, and cos zero is one. So, one times one times one: i cap dot i cap is one. Same for j cap and k cap. Clear? Now, i cap and j cap are at ninety degrees. So what is i cap dot j cap? Zero, yes! Cos ninety is zero. So any two different unit vectors give zero. Right? Now write both vectors in components and multiply out A dot B, term by term. The mixed terms vanish, and only the matching pairs survive. So, A dot B equals A sub x B sub x, plus A sub y B sub y, plus A sub z B sub z. The small x, y and z below are subscripts: A sub x is the x component of A vector. That's it! Two quick results. First, dot a vector with itself: the angle is zero, so A dot A equals A squared, the length squared; the small two on top means squared. Second, A dot B equals B dot A: order does not matter. Our second example needs both.

3:03The angle, and the perpendicular test

Now turn the formula around: cos theta equals A dot B, over A B, right? Components give the dot product, and the dot product gives the angle. That is how we will crack our room puzzle. Now suppose A dot B comes out zero. With ordinary numbers, a product is zero only when one factor is zero. So one of the vectors must be the zero vector, right? No, no, careful! Look at A B cos theta. If both lengths are non-zero, cos theta must be zero, so theta is ninety degrees. The vectors are perpendicular! The double arrow on screen means, if and only if: perpendicular if and only if A dot B is zero. And note the condition in brackets: non-zero vectors. This perpendicular test comes up often. Clear?

3:54Component of A along B

Back to the picture. That shadow, A cos theta, has a name: the component, or projection, of A vector along B vector. To get it from the dot product, do we divide A dot B by A, or by B? By the length of B vector! See, A dot B is A B cos theta, and we want only A cos theta. So the component of A along B equals A dot B, over B, got it? Dividing by the wrong length is a common slip. So divide by the length of the vector you project onto.

4:26The physics: work and power

Now, the physics. Work: W equals F vector dot s vector, where F vector is the force and s vector is the displacement. The dot product keeps only the part of the force along the displacement. With me? So tell me: if the force is at right angles to the displacement, how much work does it do? Zero! Correct! At right angles, the dot product is zero, so W equals zero. On screen, the upside-down T means perpendicular, and the arrow pointing right means implies, okay? Power is the same idea: P equals F vector dot v vector, where v vector is the velocity. Work and power: both scalars, both dot products. Clear?

5:12Example 1: angle between i + j and j + k

Now, our room puzzle. Put the x and y axes along the two floor edges, and the z axis up the corner. Then the floor diagonal is i cap plus j cap, and the wall diagonal is j cap plus k cap. Find the angle between them. Pause the video, and try it first. Done? Okay, let's check. The brackets hold each vector together. The components are one, one, zero, and zero, one, one. Multiply matching pairs: zero, plus one, plus zero. So the dot product is one. Now, the dot product is one. So cos theta is one, and the angle is zero, right? No, no, wait! We forgot the lengths. The straight bars mean magnitude: mod of i cap plus j cap. By A dot A equals A squared, one plus one is two, so each length is root two; that is the root sign. Clear? So, cos theta is one, over root two times root two: one half. So theta is sixty degrees; the little circle is the degree sign. The answer to our puzzle: sixty degrees! Not forty five, not ninety. No protractor needed.

6:24Example 2: |A| = |B| = |A + B|, find the angle

Now a harder one, a classic JEE style question. Mod A vector equals mod B vector equals mod of A plus B vector. So both vectors, and their sum, have the same length. Find the angle between A vector and B vector. Pause and try. Hint: square the lengths. Done? Okay, let's check. We only know lengths, so use A dot A equals A squared. The length of A plus B, squared, is A plus B dotted with itself, okay? Now expand it. Many students write A squared plus B squared, like Pythagoras, and stop. Is that right? No, no, careful! Pythagoras works only for perpendicular vectors. Expand all four terms: A dot A, B dot B, and A dot B twice, since B dot A equals A dot B, clear? So we get A squared plus B squared plus two A dot B. Now, all three lengths are equal, so call each one capital A, with me? The left side becomes A squared, B squared is also A squared, and two A dot B is two A squared cos theta. So, A squared equals two A squared plus two A squared cos theta. Now divide everything by A squared. What is cos theta? One equals two plus two cos theta, so cos theta is minus one half, and theta is one hundred and twenty degrees. Watch that minus sign: drop it, and you get sixty degrees, which is wrong. Done in four lines!

7:58Quick revision: dot product, angle, projection, work

So, quick revision. The dot product is A B cos theta, or multiply matching components and add; the answer is a scalar. Zero dot product, for non-zero vectors, means perpendicular. For the component along B vector, divide by its length; and work and power are both dot products. Now go and try the practice problems on this page. You can do them now. All the best!