Components and Unit Vectors

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Here is a vector in space. It makes sixty degrees with the x axis, and sixty degrees with the y axis. What angle does it make with the z axis? Sixty again, by symmetry? Hold that thought. Today you will see that sixty is impossible, and that there are two correct answers. Our tools: components and unit vectors. In JEE physics, many problems start by breaking a vector into components. So, let's begin.

0:33Breaking a vector into two components

So, look at the left half of the picture. The blue arrow is A vector: the letter A with a small arrow on top. It makes an angle theta, this Greek letter, with the x axis. Drop dashed lines from its tip to the axes, and you get the two orange arrows: its components along x and along y, okay? Now look at the orange arrow along the x axis. In this right triangle, is it the side next to theta, or the side opposite theta? Yes, exactly! It is the side next to theta, and the side next to theta, over the slanted side, gives cos. So its length is A cos theta, just like the grey label under it says. We write it as A sub x; the small x below is a subscript, naming the axis. And plain A, without the arrow, is the length of the vector, its magnitude. Right? Now the orange arrow along y matches the dashed side opposite theta, so A sub y equals A sin theta, the grey label on the left. But careful: this needs theta measured from the x axis, like here. Measure it from the y axis, and sin and cos swap places. So always check where the angle starts. Now let's go backwards. Suppose you know only A sub x and A sub y. How long is the vector? Just add the two, right? No, no, careful! The two components are at right angles, so this is Pythagoras. So, A equals the square root of A sub x squared plus A sub y squared. The tick-shaped sign is the square root, and the small two up top means squared. Clear? Now the direction: opposite over adjacent. So, tan theta equals A sub y divided by A sub x. Two numbers give us both the length and the angle. With me?

2:18Unit vectors and three dimensions

Now the right half: three axes, x, y and z. Along them sit three short green arrows: i cap, j cap and k cap. The little hat on top, the cap, marks a unit vector: its length is exactly one, okay? So, i cap points along x, j cap along y, and k cap along z. Quick check. i cap has length one. So what is three i cap? Yes, exactly! An arrow of length three, along x. Now the blue arrow is A vector again, in space. Go A sub x units along i cap, A sub y along j cap, A sub z along k cap, and add. So, A vector equals A sub x i cap, plus A sub y j cap, plus A sub z k cap: the blue label in the picture. Got it? Now its length: Pythagoras again, in three dimensions. The straight bars mean magnitude; we read them as mod A vector, the same as plain A, okay? It equals the square root of A sub x squared plus A sub y squared plus A sub z squared. Remember from our first lesson: divide any nonzero vector by its own length, and you get the unit vector along it. So, A cap equals A vector divided by its length. Same direction, length one!

3:34Direction cosines

Now, a vector in space makes an angle with each axis. We name them with three Greek letters: alpha with the x axis, beta with y, and gamma with z. Okay? In the flat picture, the component along x was A cos theta, with theta measured from the x axis. Same idea here, with alpha. So, cos alpha equals what? Correct! It is A sub x over A, the length. In the same way, cos beta is A sub y over A, and cos gamma is A sub z over A, okay? These three are the direction cosines of the vector. Quick experiment. Square all three direction cosines, and add them. What do you get? One! Always one. The squares of the components add up to A squared, which cancels the A squared below. So, cos squared alpha plus cos squared beta plus cos squared gamma equals one; the small two means squared. So the three angles are not free. Remember our puzzle? This line cracks it.

4:38Algebra, component by component

One last idea, and it is really simple. Take a second vector, B vector, with components B sub x, B sub y and B sub z. To add, add the matching components: i cap with i cap, j cap with j cap, and so on. To subtract, subtract the same way; that is the plus-minus sign. No triangles, no angles! And to multiply a vector by a number, call it lambda, this Greek letter, just multiply every component by lambda. That's it! Got it?

5:09Example 1: resolve a 10 N force at 30 degrees

So, example one. Resolve a force of ten newtons, at thirty degrees above the x axis, into components. The capital N is the newton, the unit of force, and the small circle means degrees. Pause the video, and try it first. Done? Okay, let's check. The angle is measured from the x axis. So which component gets the cos: F sub x, or F sub y? Yes, F sub x! So, F sub x equals ten cos thirty degrees. Cos thirty is root three over two, and the dot here means multiply. So, ten times root three over two: five root three newtons. Now F sub y is ten sin thirty degrees, and sin thirty is one half. So, ten times one half: five newtons. Easy, right? Put them together with the unit vectors. F vector equals five root three i cap, plus five j cap, in newtons. That's our answer. Topper's habit: always check. Five root three, squared, is seventy five, and five squared is twenty five. Their sum is one hundred, and its root is ten newtons, our original force. Perfect!

6:24Example 2: angle with z-axis when two angles are 60 degrees

Now, our puzzle from the start. The vector makes sixty degrees with the x axis and with the y axis. What angle can it make with the z axis? Pause and try. Hint: use the direction cosines. Done? Okay, let's check. Call the angle with the z axis gamma. The squared direction cosines add up to one, so: cos squared sixty, plus cos squared sixty, plus cos squared gamma, equals one. Cos sixty is half, so cos squared sixty is one quarter. Quarter plus quarter is half, and this double arrow means it follows that cos squared gamma is one half. If gamma were sixty, cos squared gamma would be one quarter, not one half. So our guess of sixty? Impossible! Now take the square root, and gamma comes out as forty five degrees. Done, right? No, no, wait! Every square root comes with two signs. So, cos gamma equals plus or minus one over root two; that is the plus-minus sign again. Many students drop the minus, and lose half the answer. Plus gives forty five degrees, and minus gives one hundred and thirty five degrees. Both are possible: the vector can point above the x y plane, or below it. That is why the question asks what angle it can make. Puzzle solved: not sixty, but forty five or one hundred and thirty five degrees!

7:52Quick revision: components, unit vectors, direction cosines

So, quick revision. One: with theta from the x axis, A sub x is A cos theta, A sub y is A sin theta, and the length comes from Pythagoras. Two: write vectors with i cap, j cap and k cap, and work component by component. Three: the squared direction cosines add up to one, and keep both signs of the root. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!