The Cross Product

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle first. Take three points: one on the x axis, one on the y axis and one on the z axis, each one unit from the origin. Join them, and you get a triangle tilted in space. What is its area? No base on the floor, no height given. Tricky, right? Today's tool, the cross product, cracks it in a few lines. And in JEE, torque and angular momentum are both built on it. So, let's begin.

0:34Size and direction of A cross B

So, look at the right half of the picture. Here, A vector, the letter with a small arrow on top, lies along the bottom, and B vector goes up at an angle theta, this Greek letter. On the left is last time's dot product, A dot B, with the small dot in the middle: it gave A B cos theta. On the right, the label with the times sign, we read as A cross B, okay? And the cross product is all about the shaded parallelogram. Its base is A vector, okay? So tell me, how high is the tip of B vector above A vector? Yes, exactly! B sin theta. So, base into height: the size of A cross B is A B sin theta. Dot had cos, cross has sin. The two straight bars mean magnitude, so we read it as mod A cross B, okay? Plain A and plain B, without arrows, are the lengths. Clear? But a cross product is a vector, so it needs a direction. Imagine the shaded parallelogram lying flat on a table. Now look at the green arrow marked A cross B, okay? Does it lie in the plane of the two vectors, or not? Absolutely, it stands straight up out of that plane, like a pole on the table! So, A cross B is perpendicular to both A vector and B vector, okay? But two directions are perpendicular to that plane: up and down. To choose, we use the right-hand rule, written in the corner. Curl the fingers of your right hand from A vector to B vector; your thumb gives A cross B, okay? Here, it points up. Try it with your own hand! Remember the shaded parallelogram, with A vector and B vector as two sides? Its area, base into height, is exactly mod A cross B, okay? And the triangle on the same two sides is half of it. See, this half is the key to our tilted triangle!

2:24Calculating A cross B from components

Alright, now let's calculate it from components. We write a three by three determinant; these two vertical bars mean determinant, okay? Top row: i cap, j cap, k cap, the letters with little caps, our unit vectors. Middle row: A sub x, A sub y, A sub z, the components of A vector. Bottom row: the components of B vector. With me? Now expand along the top row. For i cap, cover its row and column, multiply the four numbers left along the two diagonals, and subtract. That fills the bracket, shown here with dots. Same for j cap and k cap, with the signs plus, minus, plus. That minus in front of j cap is where many students lose marks!

3:08Unit vectors and the order trap

Now the unit vectors. i cap cross j cap is k cap, j cap cross k cap is i cap, and k cap cross i cap is j cap. See the pattern? i, j, k, then back to i. Each one crossed with the next gives the third. Simple as that. With ordinary numbers, the order of multiplication does not matter, does it? So j cap cross i cap is also k cap, right? No, no, careful! This is a classic trap. Swapping the order flips the direction. So, B cross A equals minus of A cross B: same size, opposite direction. So, j cap cross i cap is minus k cap. Forward in the pattern gives plus; backward gives minus. Got it? Now cross a vector with itself. The angle is zero, and sin zero is zero. So, A cross A is the zero vector, written as a zero with an arrow on top, okay? Same for any two parallel vectors: their parallelogram is squashed flat, with zero area. This double arrow, pointing both ways, means if and only if. So, two vectors are parallel if and only if their cross product is the zero vector.

4:17One identity, and the physics

Now one useful identity, handy in JEE problems. Mod A cross B, whole squared, plus A dot B, whole squared, equals A squared B squared. The dot is last time's dot product. Why is this always true? Think about sin and cos. Yes, sin squared plus cos squared is one! Square A B sin theta and A B cos theta, and add. The sin squared and cos squared add to one, leaving A squared B squared. That's it! And this is where physics comes in. Torque, the Greek letter tau, is r vector cross F vector: the position of the point where the force acts, crossed with the force. And angular momentum, L vector, is r vector cross p vector, where p vector is the momentum. Keep the order: r first. Okay?

5:08Example 1: (2i + 3j) cross (i - j + 2k) by determinant

Now, example one. Find the cross product of two i cap plus three j cap, with i cap minus j cap plus two k cap. Pause the video, and try it first. Done? Okay, let's check. The two vectors become the rows of the determinant. But look at the first vector: it has no k cap part. So what do we write in its place? Zero, yes! Never leave a gap. So the rows are two, three, zero, and one, minus one, two. Clear? Now i cap: cover its row and column. Three into two, minus zero; the small dot means multiply. So, six i cap. Next, the j cap term. The i cap term was plus, so this one is plus too, right? No, no! The signs go plus, minus, plus, so j cap always carries a minus. Inside: two into two, minus zero, that is four. So, minus four j cap. Last, k cap, with a plus sign. Two into minus one, minus three into one: minus two minus three, so minus five k cap. So the answer is six i cap, minus four j cap, minus five k cap. Not bad, is it? But how do we know it is right? Topper's trick: the answer must be perpendicular to both vectors, so its dot product with each must be zero. With the first: twelve minus twelve plus zero, zero. With the second: six plus four minus ten, zero again. Correct! Perfect.

6:37Example 2: area of a triangle from its three vertices

Now, our puzzle from the start. Find the area of the triangle with vertices one, zero, zero; zero, one, zero; and zero, zero, one. Pause and try it. Hint: write two sides as vectors. Done? Okay, let's check. Start at one, zero, zero. Going to zero, one, zero, the side is minus i cap plus j cap. Going to zero, zero, one, the side is minus i cap plus k cap. Now expand the determinant, with these two sides as its rows. For i cap: one into one, minus zero. Then minus j cap times, minus one into one, minus zero. And k cap: zero, minus one into minus one. Careful with the double minus: it makes the j cap term plus. So the cross product is i cap plus j cap plus k cap. Its magnitude, with the tick-shaped root sign, is root of one plus one plus one: root three. So the area of our triangle is root three, right? No, no, careful! Root three is the area of the parallelogram on these two sides. The triangle is half of it, so the area is root three by two. Look, our tilted triangle, solved in four lines. No base, no height needed!

7:56Quick revision: magnitude, direction, triangle area

So, quick revision. One: mod A cross B is A B sin theta, perpendicular to both vectors, by the right-hand rule. Two: expand the determinant with signs plus, minus, plus, and swapping the order flips the sign. Three: the parallelogram area is mod A cross B, and the triangle is half of it. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!