 we're now going to continue on looking at vector relations in this segment what we're going to do we're going to take a look at the dot and the cross product so let's begin with the dot product and dot product is also sometimes called the direction cosine and what the dot product does is it takes two vectors and converts those two vectors into a scalar so we're turning back to our two vectors and B so we would say a dot B sorry and we can express the dot product in the following way so that is the dot product and what we're doing here we would take the magnitude of a multiplied by the magnitude of B multiplied by cos theta and I'll draw a picture in a moment but theta is the angle between the two vectors a and B so let's take an example one that we often use in fluid mechanics and that is mass flow through a surface so I'll begin by drawing a schematic of what we're looking at so we have some area a and becoming out of that area we will have a velocity vector and I'll draw it as B and we also have a unit normal the unit normal is perpendicular to the surface a and we'll give that the symbol or the sign little n with a hat to denote the fact that it is a vector but it's a unit vector so it has magnitude of one and so we can re-express the area a as being the scalar value of area a multiplied by that unit normal vector and that would give us a representation for area a as a vector now what we can do at the dot product I mentioned it enables us to determine the mass flow rate through a surface so oh one thing that I forgot to mention here is the angle there's an angle and that is shown as being theta and it's the angle between our unit vector n and the vector v that we're interested in so we know from our analysis and fluid mechanics that the mass flux or the mass flow through that surface is going to be equal to the density multiplied by the velocity normal to that surface and then multiplied by the a or the area so that would give us something in kilograms per second and that is what we would call mass flux now we can express this in terms of the dot product by writing out in the following manner that would then be equivalent to rho times v dot a and expanding that using the definition that we provided for the dot product would be like that and that is why it's called the direction cosine because what you're doing is you're projecting a vector v in the direction of n and being this direction right here to projecting v into the direction of n giving you the normal velocity enabling you then to evaluate the mass flow through the surface so that is dot product we use that quite often in fluid mechanics and the next one we're going to look at is cross product so the cross product what it does is it takes two vectors and it produces one vector and we'll use it quite a bit when we're computing vorticity and we'll see that later on in the course but the cross product again dealing with our two vectors a and b this would read a cross b and the definition is that's it written out as a matrix the vector operator and I'll show you that in a moment but the other way would be the magnitude of a multiplied by the magnitude of b and then we have sine theta remember the dot product had a cos theta this has a sine theta and it is operating in the direction e that we do not yet know but we use the thing called the right hand rule in order to enable us to figure out what that direction e is and so before we get to the right hand rule let me draw a little picture of these two vectors so let's say we have vector a here vector b over here and I'm going to draw this angle theta between those two vectors now with the right hand rule what you do is you take your right hand and you point it at your vectors so you have your fingers curl in the direction from vector a to vector b whoops that should be a vector so put your right hand there have your vectors curl curl your fingers and in the picture curl your fingers from a to b and your thumb is going to point in the direction of e I'm sure you've seen this in physics but that is how to determine the direction of e that comes out of our cross product equation and we can also express the cross product and and this is another way of expressing it and this is the one that we'll use quite often as well you start with the i component and going through the operation we will have a y e z minus a z b y the middle you always subtract j it's not plus it's a minus j I always draw lines on the on the matrix here in order to figure out how it is but everybody has their own technique of doing this and then it's plus in the k direction which is going to be a x b y minus a y b x hopefully I didn't make a mistake there that's supposed to be an ax okay that looks good so that is the dot product and the cross product and we'll be using dot product mainly for mass flux and cross product when we look at vorticity which represents the circulation in a flow field and we'll look at that later on in the course so those are two very important vector operators that we'll be using