Part C: Concept Checks

Transcript

0:00Introduction

Hello everyone! Here is a quick test. With numbers, if two times b equals two times c, you simply cancel the two, and b equals c, right? Now take vectors: a cross b equals a cross c, with vector a not zero. Can we cancel a, and say b equals c? Most students say yes at once. Hold that thought. Today we do six quick concept checks from Part C of our vector algebra problem bank; checks like these come up often in JEE questions. So, let's begin.

0:35When does a dot b equal mod a times mod b?

So, check one is about the dot product. The small arrow on top marks a vector. Vector a dot b equals mod a, times mod b, times cos theta; the vertical bars give the modulus, the length. Theta, the Greek letter, is the angle between them, from zero up to pi; the little signs mean less than or equal to. Okay? Now the course asks: can a dot b equal mod a times mod b for unlike parallel vectors, pointing in opposite directions? They are parallel, so surely yes, right? Hmm, wait, let us check carefully. For equality, cos theta must be one, so theta is zero. That means like parallel vectors, pointing the same way, or a zero vector, which makes both sides zero. Clear? But for unlike parallel vectors, theta is pi, so a dot b is minus mod a times mod b. So the answer is no, careful! Parallel is not enough; the direction matters too. Got it?

1:39Dot product and cross product both zero

Next check. Suppose a dot b is zero, and a cross b, with the cross sign, is the zero vector, the zero with an arrow on top. The dot product gives mod a, times mod b, times cos theta, equals zero. Okay? The cross product has length mod a, times mod b, times sine theta, so that is zero too. Right? Now let me ask you this: can cos theta and sine theta both be zero for the same theta? That is right, they never can! Cos squared theta plus sine squared theta is always one, so the two are never zero together; the small two means squared. With me? So mod a times mod b must be zero, and one of the vectors is the zero vector. Keep this in mind for Example two. Clear?

2:27Swapping two vectors in a scalar triple product

Now the scalar triple product: a dot, b cross c in round brackets, which we write as a b c in square brackets. In the figure, vectors a, b and c from the corner O build a slanted box, a parallelepiped. The orange arrow is b cross c, the dashed green line h is the height, and the label says the volume is the modulus of a b c. Okay? Move the vectors round in a cycle, a b c, then b c a, then c a b, and the value stays the same. Right? Now, is a b c equal to b a c? Same three vectors, same box, so the same value, right? Careful, the answer is no! The order b a c is not a cycle; we only swapped a and b, and swapping two vectors changes the sign. Same volume, opposite sign. Clear? So the two are equal only when the value is zero, that is, when a, b and c are coplanar, lying in one plane. With me?

3:28When are the projections of a on b and b on a equal?

Now projections. In the figure, vectors a and b from O make an angle theta. Drop a perpendicular from the tip of a onto b; it meets b at N, where the small square marks a right angle. The orange arrow, O to N, shows the projection of a on b; the green arrow is the part of a perpendicular to b. Under the brace, its length is mod a cos theta, or a dot b hat; the cap, called a hat, marks the unit vector along b. So the projection, this length O N, a number with a sign, is a dot b, over the modulus of b. Okay? Turn it round, and the projection of b on a is b dot a, which is a dot b, over the modulus of a. Right? So, when are these two projections equal? The tops are the same, a dot b. So, first case: equal bottoms, mod a equals mod b. Clear? Second case, the one students forget: the top is zero, and both projections are zero. And a dot b equal to zero means a is perpendicular to b; the upside down T means perpendicular. Got it?

4:40Example 1: is the cross product associative?

Now Example one. Is the cross product associative? Is a cross b, then cross c, always equal to a cross, b cross c? Here is a hint: test it with i hat, j hat and j hat, the unit vectors along the axes. Pause the video and try it first. Done? Okay, let's check. First, i hat cross j hat is k hat, from the cycle i, j, k. So, on the left side, this k hat is then crossed with j hat. What is k hat cross j hat? Yes, it is minus i hat! We know j hat cross k hat is i hat, and swapping the order of a cross product flips the sign. Right? Now the other side. Here, j hat cross j hat is the zero vector, since the angle between them is zero, and i hat cross the zero vector is again the zero vector. Clear? Minus i hat is not the zero vector; the slashed equals sign means not equal. So the cross product is not associative, and one counterexample is enough. Always keep the brackets. Got it?

5:51Example 2: does a x b = a x c with a nonzero give b = c?

Now Example two, our puzzle from the start. Does a cross b equal to a cross c, with vector a not zero, imply that b equals c? Pause the video and try it first. Done? Okay, let's check. Bring everything to one side, and take vector a out as a common factor; the double arrow means implies. So a cross, b minus c, is the zero vector. Okay? Vector a is not zero, so b minus c must be zero, and b equals c, right? Oh no, careful, that is the trap! The cross product is also zero for parallel vectors; the two vertical lines mean parallel. So b minus c is only parallel to a, and b equals c plus t times a, for some number t. So b need not equal c, and we cannot cancel a. Clear? But suppose we also know that a dot b equals a dot c. Then, in the same way, a dot, b minus c, is zero. Right? So b minus c has a zero cross product and a zero dot product with a. Which earlier check does this remind you of? Yes, exactly, our second check! Vector b minus c is both parallel and perpendicular to a, and a is not zero. With me? So one of them is the zero vector, and since a is not zero, b minus c is zero, so b equals c. To cancel a, you need both conditions, cross and dot. That answers our puzzle! Got it?

7:27Quick revision: dot, cross and triple product traps

So, quick revision. Vector a dot b equals mod a times mod b only for like parallel vectors, or a zero vector. Zero dot product and zero cross product together mean one vector is zero. Swapping two vectors flips the sign of the scalar triple product; cyclic shifts keep it. The projections are equal when the lengths are equal, or the vectors are perpendicular. The cross product is not associative, and you cannot cancel from it, unless both the cross and the dot conditions hold, with a not zero; then b equals c. Well done on all six checks! Now try the practice problem on this page. All the best!