Coordinates, Distance and Section Formulae

Transcript

0:00Introduction

Hello everyone! Let us start with a puzzle. There is a fly in your classroom, and you want to tell a friend exactly where it is. So you say, two metres from the front wall, and three metres from the side wall. But is that enough? The fly could be near the floor, or up near the ceiling, right? We need a third number, the height. So in space, every point needs three numbers. Today we use these three numbers to find distances and dividing points, and this comes up often in JEE exams. So, let's begin.

0:34Coordinates in space

So, take three axes, x, y and z, through one point O, called the origin. They are mutually perpendicular: each is at a right angle to the other two. Okay? Any point P is written as P, with x comma y comma z in round brackets. These are signed distances, so they can be positive or negative. Right? The y axis and the z axis together make the y z plane. So tell me, the x coordinate of P is its distance from which plane? It is the y z plane, yes, exactly! That is the one plane without the x axis in it. So x is measured from the y z plane, y from the z x plane, and z from the x y plane. Clear? In a plane, two axes make four quadrants, right? In space, the three coordinate planes make eight parts, and these are called octants. Got it?

1:32Distance between two points

Now, distance. In space, we just add one more term, for z. P Q is the square root of, x two minus x one, the whole squared, plus y two minus y one, the whole squared, plus z two minus z one, the whole squared. The small numbers written low are subscripts for the two points, the raised two means squared, and the tick is the square root sign. Okay? From the origin, the first point is zero, zero, zero, so O P is the square root of x squared plus y squared plus z squared. With me? Let's try one from the course: one, two, minus three, and four, minus two, nine. Work it out quickly. The differences are three, minus four, and twelve, and squaring removes the minus sign. Be careful, nine minus minus three is twelve, not six. Nine plus sixteen plus one hundred and forty four gives one hundred and sixty nine, and its square root is thirteen. Clear?

2:34Distance from an axis and from a plane

Now a common trap. How far is the point P from the x axis? Many students say it is just x. Is that correct? No, that is wrong, and many students lose marks here. Sliding P along the x direction does not change its distance from the x axis. Only y and z count, so the distance is the square root of y squared plus z squared. Okay? Next, the distance from the x y plane, the floor, where z is the height. So the distance is mod z, z between two vertical bars, the modulus of z. Below the plane, z is negative, but a distance never is. Got it?

3:18Section formula in space

Now the section formula. Say R lies between P and Q, and divides P Q internally in the ratio m is to n. The colon is read as, is to. Does m multiply the coordinates of P, or of Q? It multiplies the coordinates of Q, a cross pairing. So the x coordinate of R is m times x two, plus n times x one, over m plus n, and y and z follow the same pattern. Right? For external division, R lies on the line P Q but outside the segment. Just replace n by minus n. So a negative ratio means external division. Clear?

3:58Topper's trick: where a coordinate plane cuts

Now a trick that toppers use. Suppose the y z plane cuts the join of P and Q, okay? What is x for every point on that plane? It is zero, yes, exactly! So in the section formula, the top of the x coordinate is zero: m times x two, plus n times x one, equals zero. With me? Solve it, and m is to n equals minus x one is to x two. One line, and the ratio is ready! The other coordinate planes work the same way.

4:31Centroid and collinear points

Okay? Two more quick results. The centroid of a triangle is the average of its three vertices: add the three x coordinates and divide by three, and the same for y and z. Okay? And a tetrahedron has four vertices, so its centroid is the average of all four. Simple, right? Last, A, B and C are collinear, on one line, if the vectors A B and A C are proportional, one a fixed multiple of the other. The arrow on top marks a vector. Clear?

5:02Example 1: show A, B, C form a right isosceles triangle

Alright, Example one, from the course. Show that A, zero, seven, minus ten, then B, one, six, minus six, and C, four, nine, minus six, form a right angled isosceles triangle. Here is a hint: compare the squares of the sides. Pause the video and try it first. Done? Okay, let's check. For A B, the differences are one, minus one and four, so A B squared is one plus one plus sixteen, which is eighteen. For B C, they are three, three and zero, so B C squared is nine plus nine plus zero, eighteen again. Right? For C A, the differences are minus four, minus two and minus four, so C A squared is sixteen plus four plus sixteen, which is thirty six. Okay? So A B equals B C, and the triangle is isosceles. And eighteen plus eighteen is thirty six, so Pythagoras holds. Now, where is the right angle, at A, at B, or at C? The right angle is at B, opposite the longest side, C A, okay? And see, we never took a square root; the squares were enough.

6:20Example 2: ratio and point where the yz-plane cuts a join

Got it? Now Example two, a little harder. In what ratio does the y z plane divide the join of minus two, four, seven, and three, minus five, eight? And find the point. Pause the video and try it first. Done? Then here is a trap first. Some students write minus x one is to x two as minus two is to three, and call the division external. Is that right? That is wrong, so be careful. Let the ratio be k is to one. The x coordinate of the dividing point is three k minus two, over k plus one, and on the y z plane this is zero. Okay? So k is two thirds. It is positive, so the division is internal, in the ratio two is to three. The shortcut agrees, because x one is minus two, so minus x one is plus two. The double minus was the trap. Got it? Now the point, with m as two and n as three. For y, two times minus five, plus three times four, over five, gives two over five. For z, two times eight, plus three times seven, over five, gives thirty seven over five. With me? So the point is zero, two over five, thirty seven over five. Its x coordinate is zero, so it really lies on the y z plane. Okay?

7:45Quick revision: 3D distance, axis distance, section ratio

So, quick revision. Distance in space is the same formula, with one more squared difference, for z. From the x axis, the distance is the square root of y squared plus z squared, and from the x y plane, it is mod z. In the section formula, m goes with Q, and the y z plane cuts the join in the ratio minus x one is to x two. Now go and try the practice problems on this page. All the best!