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1 Geometric Definition of Dot Product 1.2 The Dot Product Suppose you are pulling up on a rope attached to a box, as shown above. How would you find the force moving the box towards you? As stated above, it is the projection of u onto v. Now, supposing the force vector, F, was displacing the object along v, then the projection of F onto v would be given by the length of the vector, F, times the cosine of the angle between the two vectors. If you need to see this a little more clearly, remember that cosine is defined to be the ratio of the adjacent side to the hypotenuse; proj cos( ) v F F where proj v F is the projection of F onto v. Therefore, geometrically, we would like to define the dot product of F and v to be F v v F cos( ) v proj v F This also helps us define the work performed by the component of F through a displacement v to be Wor k = pro j v F v v F cos( ) F v

2 The following illustrations show how projection of one vector onto another works when the angle is acute and when the angle is obtuse If we are given the vectors in polar form, the geometric definition of dot product could be easily computed. Exercises: Compute the dot product of the following vectors. 1. (3,30 ) and (2,75 ) 2. (-1,45 ) and (4, 90 )

3 Algebraic Definition of Dot Product However, if you are given the vectors in rectangular form, how would we find the angle between the two vectors? Suppose we are given two vectors, A ( A cos( ), A sin( )) and B ( B cos( ), B sin( ) ). β Then, is the angle between them and by the geometric definition of dot product, A B A B cos( ) A B cos( ) A B [cos( )cos( ) sin( )sin( )] A cos( ) B cos( ) A sin( ) B sin( ) Now if, instead, we wrote A ( a1, a2) and B ( b1, b2) then A B ab a b This is the algebraic definition of dot product. It is clear, that given two vectors, we could also use the geometric definition of dot product to calculate the angle between the vectors.

4 Geometrically, we can generalize the definition of dot product to all dimensions by considering the angle between the two vectors on the plane which they generate. Algebraically, the definition of dot product can be generalized to all dimensions by extending the sum of the products of all of the corresponding coordinates. For example in three dimensions, What would the algebraic definition of dot product in 4, 5, or n dimensions be? (4-D) (5-D) (n-d) As in two dimensions, we can also find the angle between two vectors in three or more dimensions. Exercise: 1. Compute B C 2. Find the angle between the vectors,c and B. 3. How would you find angle θ using dot product? Hint: You will have to use other vectors.

5 Properties of Dot Product Due to the algebraic and geometric nature of dot product, some interesting properties hold. We need to see why these are true. The reason for this is because when we study cross product in the next section, you will want these same properties to be true for it as well. Rather than try to memorize which properties hold for each operation, you will be able to know which property holds by thinking about the definition. If this alone does not motivate you, I ll give you a quiz in which you will have to explain why any of these given properties hold so there. Proofs of properties Let u and v be vectors. Then u v u1v1 u2v2 unvn vu 1 1 vu 2 2 vnun v u Note: We did not assume that u and v were two dimensions or three dimensions, nor did we assign the coordinates values. Is there another way we could prove this property? 2. Let u and v be vectors and c be a scalar. Then c ) ( v u v ) (u v c u unvn cuv cu v cu v cu v n n Now you try the other part of property 2.

6 3. Let u, v, and w be vectors. Then u ( v w) u v u w 4. Let u be a vector. Then 5. This is obvious from the definition of dot product. More Definitions. Exercises: 1. What can you say about u v if u and v are perpendicular? 2. How would you characterize the projection of the vector, v, onto the vector, u, only using u and v? 3. Suppose u, v and w are unit vectors which are mutually orthogonal. Calculate ( au bv cw) ( au bv cw) using only a, b and c.