Scalars, Vectors and Types of Vectors

Transcript

0:00Introduction

Hello everyone! Let us start with a small puzzle. Two friends push a box, each with a force of ten newtons. So the total force is twenty newtons, right? Hmm, but wait, what if they push from opposite sides? The box does not move at all, and the total force is zero! So how can ten plus ten give zero? It is because force has a direction, and quantities with a direction do not add like ordinary numbers. Today we meet scalars, vectors and the types of vectors, the language of the whole vector algebra unit. It comes up often in JEE exams. So, let's begin.

0:42Scalars and vectors

So, first, the scalar. We call a quantity a scalar when a single real number fixes it completely. Now think of physics. Which quantities need just one number? Mass, time, temperature, work and speed, very good! None of these points anywhere. They have a size, but no direction. Right? Now, the vector. Vectors have a magnitude, which is the size, and also a direction, and they add by the triangle law. That is exactly why our ten plus ten became zero. Clear? Now look at our scalar list. Speed is sitting right there. So let me ask you this: is velocity also a scalar? No, and be careful here, because velocity is a vector. It tells you how fast and which way, while speed only tells you how fast. That is why speed stays a scalar. Displacement, velocity, force and momentum all carry a direction, so all four are vectors. Got it?

1:41Writing a vector

Now, how do we draw a vector? It is a directed line segment. The letters A and B with an arrow on top are read as vector A B, okay? It starts at A, the initial point, and ends at B, the terminal point, so the arrow shows the direction, from A towards B, right? Its magnitude is simply its length. We put two vertical bars around vector A B, and read it as mod A B, the modulus of vector A B, or we just write A B with no arrow. Clear? Usually we name vectors with small letters and an arrow: vector a, vector b, and the three dots mean and so on. The magnitude of vector a is mod a, also written as just the letter a, with no arrow. So, with the arrow it is a vector, and without the arrow it is just a number, a length. With me?

2:31Zero vector and unit vector

Alright, now the types of vectors. First, the zero vector, a zero with an arrow on top. Its magnitude is zero, and its direction is undefined. In the brackets you see vector A A, which starts at A and ends right there. No length, so no direction. Okay? Next, the unit vector. It is any vector whose magnitude is exactly one. Right? Now take any vector a. How do we get a unit vector that points in the same direction as vector a? Divide the vector by its own length, yes, exactly! So a hat equals vector a over mod a. The little cap on top is called a hat, and it always marks a unit vector. But careful: the equals sign with a slash means not equal to, so this works only when vector a is not the zero vector. We can never divide by zero. Clear? One more fact for today. The unit vectors along the x, y and z axes are i hat, j hat and k hat. The length of a one i hat, plus a two j hat, plus a three k hat, is the square root of a one squared, plus a two squared, plus a three squared. The small numbers written low are subscripts, the small two written high means squared, and the tick shaped sign is the square root. Got it?

3:47Equal vectors and negative vectors

Now, a question for you. Two vectors have exactly the same length. Are they equal vectors? Not necessarily, and this is where many students slip. Equal vectors need the same magnitude and the same direction, both together. Same length alone is not enough. Okay? But the starting point does not matter: equal vectors may have different initial points. We work with free vectors, so you may move a vector anywhere, keeping its length and direction. Right? Next, the negative of vector a: the same magnitude, but the opposite direction. So vector B A equals minus vector A B, the same segment walked backwards, from B to A, okay?

4:30Co-initial, collinear and coplanar vectors

Now three more types, quickly. Co-initial vectors share the same initial point, so they all start together. With me? Collinear vectors, also called parallel vectors, lie along the same line or along parallel lines. In symbols, vector b equals lambda times vector a, for some scalar lambda, where vector a is not the zero vector. Lambda is the Greek letter on screen, just a real number. Right? Now think. When lambda is positive, vector b points the same way as vector a. What happens when lambda is negative? The direction flips, yes! So the vectors are called like when lambda is greater than zero, with the same direction, and unlike when lambda is less than zero, with opposite directions. In both cases they are still collinear. Clear? Next, coplanar vectors are parallel to one plane. So tell me, if we pick any two vectors at all, are they always coplanar? Yes, they always are! Move both to one starting point, since we work with free vectors, and a single plane holds them. But three vectors need not be coplanar, because the third can point right out of that plane. Got it?

5:40Example 1: unit vector along 2i - 3j + 6k

Okay, let's solve some from the course. Find the unit vector along vector a, which is two i hat, minus three j hat, plus six k hat. Pause the video and try it first. Done? Okay, let's check. Step one is the length of vector a. Some students simply add the components: two minus three plus six, which gives five. Is that the length? No, no, that is the trap, because length uses squares. Two squared is four. Minus three in brackets, squared, is plus nine. And six squared is thirty six. Four plus nine plus thirty six gives forty nine, and the square root of forty nine is seven. Okay? Now divide vector a by its length, seven. So a hat equals one seventh times, two i hat minus three j hat plus six k hat. Vector a is not the zero vector, so the division is allowed. Right?

6:37Example 2: collinear test, like or unlike vectors

Now a slightly harder one. Vector a is two i hat, minus three j hat, plus four k hat. Vector b is minus four i hat, plus six j hat, minus eight k hat. Are they collinear? And are they like or unlike? Pause the video and try it first. Done? Okay, let's check. We need vector b equals lambda times vector a, so compare the matching components. Minus four over two is minus two. Six over minus three is minus two. And minus eight over four is also minus two. All three ratios agree. Okay? So vector b equals minus two times vector a, and the two vectors are collinear. Many students stop here and say they point the same way. So, are they like or unlike? They are unlike, that is correct! Lambda is minus two, which is less than zero, so vector b points opposite to vector a. Collinear only means the same line or parallel lines; the sign of lambda decides the direction. Got it?

7:44Quick revision: unit vector and collinear vectors

So, quick revision. Scalars are fixed by one real number, while vectors have a magnitude and a direction. For a unit vector, a hat equals vector a over its modulus, but only when vector a is not the zero vector. And when vector b equals lambda times vector a, they are collinear: like for positive lambda, unlike for negative lambda. Now go and try the practice problems on this page. All the best!