Angle, Intersection and Coplanarity of Two Lines

Transcript

0:00Introduction

Hello everyone! Two straight lines in your notebook that are not parallel always meet, right? Now picture a flyover: one road on top, another road underneath, going a different way. Not parallel, and yet they never meet. Strange, isn't it? In space, two lines get a third option. Today we find the angle between two lines, and learn to tell whether they are parallel, meeting, or skew like those two roads. This comes up often in JEE, so let's begin.

0:29Two lines in space

First, our two lines. L one is r vector equals a one, plus lambda times b one, and L two is r vector equals a two, plus mu times b two. The arrows on top mean vectors, the small numbers below are subscripts, and lambda and mu, the Greek letters, are parameters. In each line, vector a gives a point, vector b the direction. Okay?

0:52The angle between two lines

Now, the angle between the two lines. Let me ask you this: out of a one, b one, a two and b two, which vectors decide the angle? Yes, exactly, only the directions, b one and b two! Slide a line parallel to itself and the angle does not change, so the points do not matter. Right? So, cos theta equals the modulus of b one dot b two, over the modulus of b one times the modulus of b two. Theta is the angle, the vertical bars are the modulus, and the dot is the dot product. So why the modulus on top? Because a line has no front and no back: its direction can be b one or minus b one, so the dot product could be negative. The modulus stops that, so theta is at least zero and at most ninety degrees; the small circle means degrees. Clear? And for perpendicular lines, theta is ninety degrees and cos theta is zero, so b one dot b two equals zero. Got it?

1:54Parallel, or the same line?

Next, parallel lines. Two lines are parallel when b one is parallel to b two; the two small vertical lines between them mean parallel. In other words, their direction ratios are proportional. Right? But careful, parallel directions are not the full story. If the point a two also lies on the first line, the lines are coincident, the same line written twice. With me?

2:19Coplanar, intersecting or skew?

Now the most important test of today. Two lines are coplanar, lying in one plane, if and only if a two minus a one, dot b one cross b two, equals zero. The double arrow means if and only if, the cross is the cross product, and this is the scalar triple product. Okay? But why does it work? The vector a two minus a one joins a point of L one to a point of L two. If both lines lie in one plane, where do this vector, b one and b two lie? In that same plane, exactly! Three vectors in one plane enclose zero volume, so their scalar triple product is zero. Clear? In coordinates, it is this determinant; the vertical bars here mean a determinant. The first row is x two minus x one, y two minus y one, z two minus z one, from the two points. The other rows, a one, b one, c one and a two, b two, c two, are the direction ratios of the two lines, plain numbers, not vectors. Okay? Two lines in one plane are either parallel or they meet. So coplanar and not parallel means they intersect. Right? And lines that are neither parallel nor intersecting are skew, like our flyover.

3:34Finding the point where they meet

Got it? So how do we find the meeting point? Step one: write the general point of each line, with t for the first and s for the second. Okay? Step two: equate them. That gives three equations, one for each coordinate, but only two unknowns. So what is the extra equation for? Yes, it is our check! Solve two of them, then put t and s into the third. Clear? If the third one fails, there is no common point; and if the lines are not parallel either, they are skew. With me?

4:09Example 1: angle between two lines in vector form

An easy one from the course first. Find the angle between r vector equals i hat plus j hat, plus lambda times i hat plus two j hat plus two k hat, and r vector equals k hat, plus mu times three i hat plus two j hat plus six k hat. The little caps are hats: i hat, j hat and k hat are the unit vectors along the axes. Pause the video and try it first. Done? Okay, let's check. Some students start with i hat plus j hat, and k hat. Do we need those? No, those are just points on the lines! Only the directions matter, so b one is one, two, two, and b two is three, two, six. Right? The dot product is three plus four plus twelve, which gives nineteen. It is already positive, so the modulus changes nothing. Okay? Now the lengths, with the square root sign. The root of one plus four plus four is three, and the root of nine plus four plus thirty six is seven. Clear? So cos theta is nineteen over three times seven, which is nineteen over twenty one; the small dot means times. Theta is cos inverse of nineteen over twenty one, where the minus one up there means inverse, not a power. And that is our answer, really simple!

5:27Example 2: show two lines intersect, find the point

Now a harder one, where our coplanarity test does the real work. Line one is x minus five over four, equals y minus seven over four, equals z plus three over minus five. Line two is x minus eight over seven, equals y minus four over one, equals z minus five over three. Show that they intersect, and find the point. Pause the video and try it first. Done? Okay, let's check. The points are five, seven, minus three and eight, four, five, so a two minus a one is three, minus three, eight. Right? Next, b one cross b two, with directions four, four, minus five and seven, one, three. Twelve plus five gives seventeen; minus thirty five minus twelve gives minus forty seven; and four minus twenty eight gives minus twenty four. Clear? Now the moment of truth. If these lines lie in one plane, what must the dot product of these two vectors be? It must be zero, yes! We get fifty one, plus one hundred and forty one, minus one hundred and ninety two. Fifty one plus one hundred and forty one gives one hundred and ninety two, so the total is zero. Coplanar, and the directions are not proportional, so they intersect! Got it? The general point of line one is five plus four times t, seven plus four times t, minus three minus five times t. For line two it is eight plus seven times s, four plus s, five plus three times s. Equate them. With me? Equating the x coordinates gives four times t minus seven times s equals three, and the y coordinates give four times t minus s equals minus three. Subtract the second equation from the first, and minus six times s equals six, so s is minus one, and then t is minus one too. The arrow means it follows. Okay? Never skip the check. For the z coordinates, minus three plus five gives two, and five minus three gives two. They match! So the point is one, three, two. For a flyover pair, this check would fail. Clear?

7:39Quick revision: angle, coplanar test, meeting point

So, quick revision. For the angle, use only the directions, with the modulus on top. Lines are coplanar when a two minus a one, dot b one cross b two, is zero. For the meeting point, solve two equations, and check the third. Now go and try the practice problems on this page. All the best!