Continuing our discussion of mathematics, now I want to start talking about vectors. And first of all we'll look at some basic definitions and operations. And then we'll do some examples. So first of all, basic definitions. A vector of course is characterized by a magnitude and direction. And the general definition diagram for vectors is given on the sketch here. So we have a right-handed Cartesian coordinate system x, y, and z with unit vectors i, j, and k in those directions respectively. So in two dimensions, more simply, in the x, y plane, let's suppose we have some vector V, as shown here. We can write the vector V as being Vxi+Vyj, where Vx and Vy are the components of the vector in the x and y directions, the scaler components. So Vx and Vy are the rectangular components of the vector where simply Vx is equal to V cosine theta. Vy is equal to V sine theta where V is the magnitude of the vector which sometimes we write like that to be more careful. But I'll just write it as V without a vector symbol. Denotes the magnitude of the vector and theta is the angle of the vector to the x axis in this case. The magnitude of the vector V is the square root of the sum of the squares of the individual components. And the angle theta is the arc tangent Vy over Vx from simple geometry. More generally in three dimensions using the right hand rule again for the coordinate system xyx, we can write the vector in three dimensions in this form. Vxi + Vyj + Vzk and again, the magnitude of the vector is the square root of the sum of the squares of the individual components. Another useful construct is the unit vector. And a unit vector is a vector of length unity or one which lies in the direction of the original vector. So in this case for example, if we have a vector V, the unit vector along this axis here denoted by n, is equal to the vector divided by the magnitude of the vector. In other words, it has a length of one. So this is convenient because we can also write our original vector in this form, that the vector is equal to the magnitude multiplied by its unit vector. And the components of the unit vector in the x, y, and z directions we can denote by ex, ey, and ez. So we can also write the unit vector In this form. And it's also useful to use the direction cosines. The direction cosines are the cosines of the angles that the vector makes with the x, y, and z axes. So, for example, in this case, theta x, here, is this angle. The angle of the vector relative to the x axis. And the direction cosines are cosine theta x, cosine theta y, etc. And these direction cosines are in turn the scale of components of the unit vector, so ex, here is equal to cosine theta x, ey is equal to cosine theta y, etc. So we can also write the unit vector in this form, m is equal to i cosine theta x, etc. We can do a number of manipulations with vectors. Most simple, of course, being addition and subtraction. So in this case, we have two vectors, A and B. Denoted by A is axi, ayj, etc. And to add those two vectors A plus B, we simply add the components in the x, y, and z directions. So the i component is just ax plus bx, the j component is ay plus by, etc. And to subtract them, we just subtract the components of the second vector. So for example, let's suppose I have two vectors, V1 and V2 as shown here. The summation of them is given by this here. So in this case V is the resultant of those two vectors or the sum of them. And we can also obtain this by vector addition, just adding the vectors head to tail. So V1 plus V2 here. And the summation of those two is, the final line, which closes the polygon, or the triangle in this case, is V, is therefore the summation, or the resultant, which you can see, is exactly the same that we get by forming the resultant here. But again, we can do this by simply adding the individual x and y components. So if V1 is given by ivx one plus jv one one etcetera then the summation of those vectors is simply the summation of the individual components vx one by vx two and the j component, vy one plus vy two. Next we conform the dot product and the dot product is a scalar product which represents the projection for example of vector V onto another vector A multiplied by the magnitude of the vector A. So in this case, let's suppose we have a vector B and a vector a,Athen the component of the vector B, in the direction of A, is just B cosine theta as shown here. And the dot product of those two vectors is the orthogonal component, in other words the projection. Of the vector B onto A. So that is equal to A dot B is AB cosine theta. And obviously this is a scale of product. It has no direction of its own. So the component of B, in the direction of A, we can conveniently form in this way, is equal to B dot n, where in this case the unit vector in the direction of A. So the dot product is a useful way of forming the component of a vector in any other arbitrary direction in this way. So, n is the unit vector which is equal to the vector A divided by the magnitude of A. In other words, the unit vector in the direction A. We can generalize this statement here, that in this way, that the dot product of two vectors, A and B, is equal to the summation of the products of the individual components, ax bx plus ay by, etc. And that is the equation given in the reference handbook right here. And we can also see from this that A dot B is equal to B dot A. We can reverse the order of the dot product. Next we have the cross product. And the cross product is a vector product which has magnitude BA where B and A are the magnitudes of the individual vectors multiplied by the sine of the angle between them. And the cross product is a vector quantity and its direction is in a plane which is perpendicular to the plane of the two individual vectors that we're forming the cross product of. So, let's suppose I have two vectors, A and B and the cross product of them is C. Then the cross product is most easily formed by forming the determinant of I J K, the unit vectors and then the components of the vector A and then the components of the vector B like this. And the magnitude of that determinant we can form in the way that we previously did it so the I component here, cross that out. It's i multiplied by aybz minus azby, is the first term. Then the j term, the second one, j component. Cross this and this out. And the sines multiply, alternate rather. So it's minus j multiplied by axbz minus azbx is this term. And similarly for the k term, except that, because of the alternating signs is plus. So, that is the general expression for the cross product of two vectors. The physical meaning of this we can see fairly easily. Let's suppose that my two vectors are shown here A and B with an angle theta in between them. Then the direction of C, is going to be perpendicular to the plane containing those two vectors. So it's going to be in this direction here. And the direction of that vector, we get from the right hand draw. So, if I curl my fingers In the direction from the first vector to the second vector, then the direction of the cross product is given by the direction that my thumb is pointing in. So in the case the cross product a, b going from a to b in this direction, from the right hand rule, the direction of the cross product c, is in the direction which I've sketched there. And physically, the magnitude of the cross product here is given by AB sine theta. But AB sine theta is simply the area of the parallelogram between those two vectors. So the magnitude of the cross-product is the area of the parallelogram formed by the two vectors from which we can see right away that if those two vectors parallel each to each other the cross-product is 0. So C, the magnitude of the vector is AB sine theta, which is the area of the parallelogram. And we can also see that we cannot readily reverse that if we go from B to A in the other direction here. So now if I go B, A cross-product in that direction, then the direction of the resulting cross product is reversed from the right handle in other words B cross A is equal to negative of A cross B. There are a number of vector identities which are summarized and listed in the handbook, which I've given here and just to repeat some of those, the dot product of parallel unit vectors, i.i, j.j, etc is equal to 1. But the dot product of orthogonal vectors or vectors which are perpendicular to each other are zero. The cross product of parallel vectors i cross i, et cetera is zero. But the cross product of orthogonal or perpendicular unit vectors is equal to, well for example, i cross j is equal to k. J x I =- k et cetera for the others. And these products are all given in the reference handbook in this section here. For the dot products we have A dot B = B dot A. We can reverse it. And if we have summations of vectors we can just expand them in this form, A dub B plus C is equal to A dot B plus A dot C. Cross products, A cross B is equal to negative of B cross A. And again, if we have parentheses A cross B cross, A cross B plus C is equal to A cross B plus A cross C etc. And the rest of the identities are given in the table here. So, that's a discussion of basic properties of vectors and in the next segment, we'll do some examples.
