Scalars, Vectors and Displacement

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. I walk three metres, and then four metres more. Now tell me, how far am I from where I started? Seven metres, right? Wait, not so fast! In physics, the answer can be five metres. Three plus four, equal to five? Hold on to that thought. And look, velocity, acceleration, force, momentum: all of them are vectors, and they are all over JEE physics. So come on, let's begin.

0:34Number and unit: always enough?

So, let's start simple. Mass, five kilograms. Time, two seconds. Temperature, three hundred kelvin. The k g, the s and the capital K on screen are just the units, okay? Each one is complete: a number and a unit. Clear? Now suppose I walk out of this classroom and tell you, my displacement is five metres. Can you find me? No, you can't! Five metres, but which way? Front, back, left or right? See, a displacement of five metres is incomplete until we say, towards the north. So some quantities need a direction too. That's it, that is the whole difference.

1:14Scalars and vectors

So now, two names. When a quantity has magnitude only, we call it a scalar. Magnitude simply means size. And scalars add by ordinary arithmetic, simple adding, okay? Examples: mass, time, distance, speed, work and energy. Each of them is just a number with a unit. Got it? Now, when a quantity has magnitude and direction, and it adds by the triangle law, we call it a vector. The triangle law, we will do properly in the next lesson. For now, remember two conditions: direction, and the triangle law, okay? Examples: displacement, velocity, acceleration, force and momentum. Notice the pairs. Distance is a scalar, but displacement is a vector. Speed is a scalar, but velocity is a vector. Keep these pairs in mind. With me?

2:09JEE trap: electric current

Alright, now the most famous trap of this chapter. Electric current flows along a wire, so it has a direction along the wire, right? So tell me, is current a vector? If you said yes, you just fell into the trap! Look, when two currents meet at a junction, they add arithmetically, simple adding. Not by the triangle law, okay? Remember our two conditions? Direction, and the triangle law. Current has the first, but not the second. So current is a scalar; the double arrow on screen means, so. Very important. Write this down. Same story with finite rotations, that means turning an object through some angle. They also have a direction, but they do not add by the triangle law, so they are not vectors either. Direction alone is not enough, clear?

3:01Kinds of vectors

Now, writing vectors. We put a small arrow on top of the letter, so this is A vector, okay? Its magnitude is written with two straight bars around it, mod A vector, or simply as plain A, without the arrow. Got it? Next, the little cap on top of the letter A, okay? That is A cap, the unit vector along A vector. And A cap equals A vector divided by A, its magnitude. Cap on top means unit vector. That's it. Two vectors are equal when they have the same magnitude and the same direction. Both, not just one, okay? And minus A vector has the same magnitude as A vector, but the opposite direction. Just flip the arrow around. Simple, right? And last, the zero vector, with magnitude zero. Keep it in mind; it comes back in our second example.

3:55Position vector and displacement

Now the most useful vector of this chapter: the position vector of a point, r vector, drawn from the origin. r vector equals x i cap, plus y j cap, plus z k cap, where x, y and z are the coordinates of the point. Here i cap, j cap and k cap are the unit vectors along the x, y and z axes. Clear? Now a particle moves from point P one to point P two; the small one and two below the P are just labels, okay? Its displacement is delta r vector, where delta means change. Delta r vector equals r two vector minus r one vector: final position minus initial position. Now think. The particle can go from P one to P two along a straight line, or along a long, curvy road. Does the displacement change? No, not at all! Look at the formula: only r one and r two, the end points. The path does not appear anywhere. But distance is the length of the actual path, so a long road means more distance, okay? So, mod delta r vector is less than or equal to the distance, because the straight line joining the end points is the shortest path, right? And here is the topper's trick: for displacement, forget the path. Just join the start point to the end point. That's it!

5:13Example 1: distance and displacement for 3 m east, 4 m north

Now, remember our puzzle from the start? Let's solve it. So, a person walks three metres east, and then four metres north. Find the distance covered and the displacement. Pause the video, and try it first. Done? Okay, let's check. Distance first. Distance is the actual path, so just add: three plus four, seven metres. Easy, isn't it? Now the displacement. Seven metres again, right? No, no, careful! Displacement looks only at the end points. East and north are at right angles, so the start, the turn and the end make a right-angled triangle. So the displacement is the square root of three squared plus four squared: five metres. There it is, three plus four equal to five! Puzzle solved. But wait, a displacement needs a direction too, remember? The angle north of east is tan inverse of four by three; the minus one on top of tan means tan inverse. That is approximately fifty three degrees; the wavy equals sign means approximately, and the small circle means degrees. So, five metres at fifty three degrees north of east. Now it is complete! And a quick check with our rule. Mod delta r vector, five metres, is less than or equal to seven metres, the distance. Absolutely correct!

6:37Example 2: can vectors of 3, 4 and 8 add up to zero

Now a slightly harder one, JEE level. Can three vectors of magnitudes three, four and eight add up to zero? What about three, four and five? Pause and try. Hint: think of a triangle. Now, many students think like this. Vectors can point in any direction, so choose the directions cleverly and any three can cancel out, right? No, no, careful! Vectors add by the triangle law: we join them head to tail. For the sum to be the zero vector, the zero with an arrow on screen, the head of the last one must come back to the tail of the first. So they must form a closed triangle. See, the zero vector is back! And three lengths can close a triangle only when each one is at most the sum of the other two. The sign on screen means, less than or equal to, okay? First set: three, four and eight. Three plus four is only seven, and eight is greater than seven; that is the greater-than sign on screen. The two short sides cannot reach across the long one, so the answer is no. Second set: three, four and five. Five is less than or equal to three plus four, that is seven, so yes, they can add up to zero. And notice, three, four, five makes a right-angled triangle, the same one as in our walk! Beautiful.

8:01Quick revision: scalars, vectors, current, displacement

So, quick revision. Scalar: magnitude only. Vector: direction too, and the triangle law. Current has a direction, but it is a scalar. And displacement depends only on the end points, so mod delta r vector is less than or equal to the distance. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!