Direction Cosines and Direction Ratios

Transcript

0:00Introduction

Hello everyone! Let us start with a small puzzle. Can a line in space make angles of thirty degrees, forty five degrees and sixty degrees with the x, y and z axes? Three nice angles, so why not, right? Keep your answer in mind, because in a few minutes we will test it with one simple rule. This topic comes up often in JEE three dimensional geometry, so let's begin.

0:28Direction cosines: three angles, three cosines

So, here is our line: an arrow from the origin O, going out to the point P, okay? It makes three angles with the positive x, y and z axes, called alpha, beta and gamma, the Greek letters in orange. Now take their cosines. l equals cos alpha, m equals cos beta, and n equals cos gamma. These three numbers are the direction cosines of the line. Simple, right? Here is the key fact. The vector l i hat, plus m j hat, plus n k hat, is a unit vector along the line; the little caps are hats, marking the unit vectors along the axes. And the last line on the figure says l squared plus m squared plus n squared equals one. So tell me, why must that be true? Because that vector is a unit vector! Its length is the square root of l squared plus m squared plus n squared, and that length is one, so the sum is one. The small two up top means squared. This is the most important line of today's lesson. Clear? Now, back to our puzzle: thirty, forty five and sixty degrees, where the little circle is the degree sign. Test them with this rule. Is it possible or not? For these angles, the cos squares are three quarters, one half and one quarter. They add to three over two, and the slashed equals sign says that is not equal to one. So no, such a line cannot exist. With me? And the same rule in another form: since sine squared is one minus cos squared, sine squared alpha plus sine squared beta plus sine squared gamma is three minus one, which is two. Got it?

2:16Direction ratios: any multiple will do

Now, direction ratios. Any three numbers a, b, c proportional to l, m and n are direction ratios of the line. Double them, triple them, they still give the same direction. Okay? So let me ask you this: must a squared plus b squared plus c squared also equal one? No, not at all, and this is a common trap. Direction ratios can be any size. So never put them straight into a formula that needs direction cosines. Right? To get back to cosines, divide each ratio by the square root of, a squared plus b squared plus c squared; that is the root sign on screen. The sign with plus on top of minus means plus or minus, because the line can point either of two opposite ways. Clear? Here is one from the course; d r means direction ratios, and d c means direction cosines. Take the ratios two, minus one, minus two. Square and add, and you get four plus one plus four, which is nine; its root is three. So the cosines are two thirds, minus one third, minus two thirds, or all with the opposite sign. And for the line through two points P and Q, the direction ratios are just the differences: x two minus x one, y two minus y one, z two minus z one. The small one and two below the letters are subscripts that name the two points, okay?

3:45The angle between two lines

Now the angle theta between two lines; theta is a Greek letter. With direction cosines, cos theta is the modulus of, l one times l two, plus m one times m two, plus n one times n two. The straight bars are the modulus sign, so theta is never more than ninety degrees. With me? With direction ratios, the top is the same kind of sum, and we divide by the two lengths: the root of the sum of squares for each line. Right? So tell me, when are the two lines perpendicular? Yes, exactly, when theta is ninety degrees, so cos theta is zero. That means a one times a two, plus b one times b two, plus c one times c two, equals zero. Clear? And parallel lines have proportional direction ratios: a one over a two equals b one over b two, equals c one over c two. Got it?

4:41Topper's trick: one linear, one quadratic

Now a topper's trick, for a type of question that comes up often. You get two equations in l, m and n, one linear and one quadratic. Step one: use the linear one to eliminate a cosine. Okay? Step two: the quadratic then factorises into two linear factors, and each factor gives one line. With me?

5:03Example 1: angle with z-axis given 45 and 60 with x and y

Alright, our first example from the course. Read it with me: a line makes angles of forty five and sixty degrees with the x and y axes. Find its angle with the z axis. Pause the video and try it first. Done? Okay, let's check. Call the unknown angle gamma. Our golden rule says cos squared forty five, plus cos squared sixty, plus cos squared gamma, equals one. Right? Cos forty five is one over root two, so its square is one half, and cos squared sixty is one quarter. So cos squared gamma is one minus one half minus one quarter, which is one quarter. So cos gamma is one half, and gamma is sixty degrees. Done, right? Oh no, careful, there is a second answer! Cos gamma is plus or minus one half; the double arrow means implies. Plus one half gives sixty degrees, and minus one half gives one hundred and twenty degrees. So the answer is sixty or one hundred and twenty degrees.

6:09Example 2: angle between lines with l+m+n=0 and 2lm+2nl-mn=0

Clear? Now a harder one, where our topper's trick does all the work. Find the angle between the lines whose direction cosines satisfy l plus m plus n equals zero, and two times l times m, plus two times n times l, minus m times n, equals zero. Pause the video and try it first. Done? Okay, let's check. The linear equation gives n equals minus, in brackets, l plus m. Put that into the quadratic: two l m, minus two l times, l plus m, plus m times, l plus m, equals zero. Okay? Now expand. Two l m and minus two l m cancel, so only one l m survives, and we get m squared, plus l times m, minus two l squared, equals zero. And this factorises: m minus l, times m plus two l, equals zero. Right? So we have two factors. How many lines does that give us? Two lines, exactly, one from each factor! First, m equals l. Then n equals minus, l plus l, which is minus two l. So the direction ratios are one, one, minus two. Clear? Second, m equals minus two l. Then n equals minus, l minus two l, which is just l. So the direction ratios are one, minus two, one. With me? Now the angle formula. On top, one minus two minus two is minus three, and its modulus is three. At the bottom, root six times root six is six. Three over six gives one half, so theta is pi over three, written with the Greek letter pi. We never needed the actual cosines; the ratios were enough. See how fast that was?

8:01Quick revision: l2+m2+n2=1, ratios to cosines, angle formula

So, quick revision. Direction cosines satisfy l squared plus m squared plus n squared equals one; direction ratios need not. To get cosines from ratios, divide by the root of, a squared plus b squared plus c squared, with a plus or minus sign. And the angle formula uses a modulus; a zero sum of products means perpendicular. Now go and try the practice problems on this page. All the best!