Vector Tools: Area, Volume and Coplanarity

Transcript

0:00Introduction

Hello everyone! Have you ever sat at a four legged table that wobbles? And a stool with three legs, does it ever wobble? It never does, right? Three points always lie in one plane, but four points may not. Today's vector tools give the area of a triangle, the volume of a tetrahedron, and a test for four points in one plane. They come up often in JEE three dimensional geometry, so let's begin.

0:29Dot product in coordinates

First, a quick warm up. Vector a has a small arrow on top, and components a one, a two, a three, with little subscripts. Now, a dot b is a one times b one, plus a two times b two, plus a three times b three. Multiply the matching components and add. Clear? It also equals the modulus of a, times the modulus of b, times cos theta. The vertical bars mean the modulus, the length, and theta, the Greek letter, is the angle between them. So what can this second form find for you? Yes, exactly, the angle between them! cos theta equals a dot b, over the modulus of a times the modulus of b. Got it?

1:14The cross product

Now the star of today, the cross product, with a small cross between a and b. It gives a vector. Its first component is a two times b three, minus a three times b two. The second is a three times b one, minus a one times b three. The third is a one times b two, minus a two times b one. See the subscripts: two three, three one, one two, going round in a circle. That is the trick to remember it, okay? And this new vector is perpendicular to both a and b. It stands straight up from both of them, right? Its length, the modulus of a cross b, is the modulus of a, times the modulus of b, times sine theta. Sine this time, not cos, so keep that in mind. Okay? So it gives the normal to any plane containing the directions a and b, and, when a and b are the directions of two lines, the direction of their common perpendicular. With me?

2:13Cross product: a quick one

Let's try a quick one from the course: one, two, three, cross four, five, six. Many students write the middle component as one times six minus three times four, which is minus six. The small dot means times. Is that right? No, no, careful, that is the classic trap! The middle one is a three times b one, minus a one times b three, so three times four minus one times six, which is six. The first and the last, twelve minus fifteen and five minus eight, are both minus three. So the answer is minus three, six, minus three, okay? Now a good habit: check it. It must be perpendicular to one, two, three, so the dot product must be zero, as cos of ninety degrees is zero. Minus three plus twelve minus nine gives zero, so we are right!

3:07Tool one: area of a triangle

Now tool one, the area of a triangle with corners A, B and C, okay? Area needs two sides and the sine of the angle between them. So which product do we use, dot or cross? The cross product, yes, its length has the sine! The arrow over A B means the vector from A to B, and likewise for A C, okay? The modulus of A B cross A C is the length of A B times the length of A C, times the sine of angle A, right? And the triangle is half of that. So the area of triangle ABC, with the small triangle symbol, is one half of that modulus. Forget the half, and you get double the area. Got it?

3:50Tool two: scalar triple product and volume

Now comes tool number two. Three vectors in square brackets give the scalar triple product, a dot, b cross c. Here b cross c is a vector, and dotting it with a gives a number, a scalar. Okay? To compute it, write the three vectors as the rows of a determinant, the d e t on screen, and expand. It is the same three by three determinant you know. Clear? Now the volume of a tetrahedron, the solid with four triangular faces and corners A, B, C and D, okay? It is one sixth of the modulus of the scalar triple product of the edges A B, A C and A D, right? Why do we need the modulus here? Yes, exactly! The determinant can come out negative, but a volume never can. So take the modulus, and divide by six, not by two. Got it?

4:44Four points in one plane

Now, back to our wobbly table. Suppose the scalar triple product of A B, A C and A D is zero, so the volume is zero. What does that tell you about the four points? Yes, exactly! The tetrahedron is flat, so all four points lie in one plane. So four points are coplanar if and only if this scalar triple product is zero; the double arrow on screen means if and only if. That is our table test. Clear?

5:14Example 1: area of triangle (1,1,1), (1,2,3), (2,3,1)

Now, Example one from the course. Find the area of the triangle with vertices one, one, one; one, two, three; and two, three, one. Pause the video and try it first. Done? Okay, let's check. Call the points A, B and C in order. Vector A B is B minus A, which is zero, one, two, and vector A C is C minus A, which is one, two, zero. With me? Now the cross product, with our circle pattern. One times zero minus two times two, then two times one minus zero times zero, then zero times two minus one times one. That gives zero minus four, two minus zero, and zero minus one, so minus four, two, minus one. Okay? Its modulus is the root of sixteen plus four plus one, which is root twenty one. And do not forget the half, so the area is root twenty one over two. Easy, right?

6:18Example 2: value of k for four coplanar points

Now a harder one, Example two. For what value of k are the points one, one, one; two, three, k; zero, one, two; and one, zero, four coplanar? Think of our table, and pause the video to try it first. Done? Okay, let's check. Subtract one, one, one from the other three points. We get one, two, k minus one; then minus one, zero, one; and zero, minus one, three, okay? Coplanar means their scalar triple product is zero, so we put them as the rows of a determinant and set it equal to zero. Right? Now expand along the first row. Quick question: which sign goes in front of the middle term, the one with the two? It is a minus, yes, the signs go plus, minus, plus. So, one times the bracket zero plus one, minus two times the bracket minus three minus zero, plus k minus one, times the bracket one minus zero, equals zero. Clear? Now simplify, and watch the double minus: minus two times minus three is plus six. So one plus six plus k minus one is k plus six, which is zero. So k equals minus six; the arrow on screen means it follows that. For this k, the four points lie in one plane, and the table stops wobbling! Got it?

7:49Quick revision: area, tetrahedron volume, coplanarity

So, quick revision. Area of a triangle is half the modulus of the cross product of two sides. Volume of a tetrahedron is one sixth of the modulus of the scalar triple product. And four points are coplanar exactly when that product is zero. Now try the two practice problems on this page. All the best!