Addition, Subtraction and Scalar Multiples

Transcript

0:00Introduction

Hello everyone! You walk three steps, turn any way you like, and walk five more. How far are you from the start? Eight steps, because three plus five is eight, right? But that is only true if you never turned. Quantities with a direction do not add like plain numbers. Today we learn to add, subtract and scale vectors. This comes up often in JEE, so let's begin.

0:27Adding vectors: triangle and parallelogram laws

So, look at the left figure. The arrow from O to A is vector a; the small arrow on top of a letter means a vector, okay? Now walk along vector a, then along vector b. Where do you end up? Yes, exactly, at the point B, along the orange arrow! An arrow on top of two letters, like O A, means the vector from O to A, right? So vector O A plus vector A B equals vector O B; that is the triangle law, head to tail. Clear? Now the right figure. Here a and b both start from O, and we complete the parallelogram. The diagonal through O is a plus b, the same answer as the triangle law. Okay? In the parallelogram, go along a then up the far side, or up b then across the top: both paths reach the same corner, right? So a plus b equals b plus a; the order does not matter.

1:23More rules, and the polygon law

Got it? Now more rules, quickly. The brackets show what we add first, but grouping does not matter: a plus b, then plus c, equals a, plus b plus c. Right? Next, the zero with an arrow on top is the zero vector, with no length; adding it changes nothing. And minus a has the length of a, but the opposite direction, so a plus, minus a, gives the zero vector. With me? Now the polygon law. Walk A to B, B to C, and so on; the three dots mean we keep going up to Y to Z, okay? Head to tail, the total is the single vector A Z, first point to last. Now tell me this: what if you go around a closed polygon, back to the start? Yes, exactly, the zero vector! So the sides of a closed polygon, taken in order, add to zero. We use this in Example two.

2:15Subtraction: add the negative

Now, subtraction is really simple. Vector a minus vector b means a plus, minus b in brackets: reverse b, then add. Look at the green dashed diagonal, from A to C, and its label says b minus a. But many students write a minus b here, because it joins the tips of a and b. So which is right, and why? Look at the arrowhead first: it points at C, so we go from A to C, right? From A, go back to O, that is minus a, then from O to C, that is plus b. So vector A C equals b minus a. Get the order wrong and the sign flips. Clear? This gives a rule you will use all the time. Now take any two points A and B, with small a and small b the vectors to them from a common origin O, okay? Careful, this B is a new point, not the corner B of our figure. By the same reasoning, vector A B equals b minus a. In words, the end point's vector minus the start point's vector: end minus start.

3:20Multiplying a vector by a scalar

Got it? Now, scalar multiples. Lambda, the Greek letter here, is a real number, a scalar. The vertical bars mean modulus, a length or a size. So the modulus of lambda times a equals the modulus of lambda, times the modulus of a. Okay? Let me ask you this: what does minus two times vector a look like? Very good, twice as long, and reversed! When lambda is greater than zero, lambda times a points the same way as a; when it is less than zero, it reverses. Right? And brackets open just like algebra. Lambda times, a plus b, equals lambda times a, plus lambda times b. With mu, another Greek letter, lambda plus mu, times a, equals lambda times a, plus mu times a. With me?

4:10How long can a sum be? The triangle inequality

Now back to our puzzle. In the left figure, the side O B is a plus b. Can its length ever be more than the other two sides put together? No, never, because the straight path is the shortest. So the modulus of a plus b is less than or equal to, the modulus of a plus the modulus of b. The less than sign with a line under it means less than or equal to. Okay? Here is the full triangle inequality. The plus or minus sign means it holds for a plus b and for a minus b, and the outer bars make the difference of the lengths never negative. So the length lies between that difference and the sum. Right? And equality on the right, for a plus b, holds only for like parallel vectors, pointing the same way. Clear?

4:57Example 1: range of the sum of vectors of lengths 3 and 5

Now, Example one, our puzzle from the start. Two vectors have magnitudes three and five. Between what limits does the magnitude of their sum lie? Pause the video and try it first. Done? Okay, let's check. Many students write eight, because three plus five is eight. Is that the whole answer? Not quite, eight is only the upper limit. By the triangle inequality, five minus three is less than or equal to the modulus of a plus b, which is less than or equal to five plus three. So it lies between two and eight, both included. Okay? We get eight for like parallel vectors, and two for unlike parallel vectors. And that solves our puzzle: the walk ends anywhere from two to eight steps from the start.

5:50Example 2: sides and diagonals of a regular hexagon

Now a harder one, Example two. In the regular hexagon ABCDEF, vector A B equals vector a, and vector B C equals vector b. Express C D, D E, E F, F A, A D and A E in terms of a and b. Pause the video and try it first. Done? Okay, let's check. Take the long diagonal from A to D, the one through the centre. Which side of the hexagon points the same way? Yes, it is side B C: the diagonal A D is parallel to B C, in the same direction. E F is parallel too, but it points the other way. The two upright lines mean parallel. It is twice as long as a side, so vector A D equals two b. Got it? Now C D, by the polygon law. Vector A B plus B C plus C D equals A D, right? So C D equals A D, minus A B, minus B C: two b, minus a, minus b, giving b minus a. Clear? Now D E, the side opposite to A B, okay? How are these two sides related? Very good, equal and parallel, but opposite in direction! So D E is minus a, and E F, opposite to B C, is minus b. And F A, opposite to C D, is minus, b minus a, so a minus b. With me? For A E, go from A to D, then from D to E: that is two b, and then minus a. So vector A E equals two b minus a. Okay? Now a topper's check. The six sides of the hexagon, in order, must add to zero. Add our answers: a, plus b, plus b minus a, minus a, minus b, plus a minus b. Every a and every b cancels, giving the zero vector. Very good work!

7:42Quick revision: add, subtract, triangle inequality

So, quick revision. To add, go head to tail, or take the diagonal of the parallelogram. To subtract, add the negative: vector A B is b minus a, end minus start. And the length of a plus or minus b lies between the difference and the sum of the lengths. Now try the practice problems on this page. All the best!