Unformatted text preview: is the scalar âˆ’â†’ v Â· âˆ’â†’ w = x 1 x 2 + y 1 y 2 + z 1 z 2 . The deÂ±nition of the dot product exhibits a number of algebraic properties, which we leave to you to verify (Exercise 3 ): Proposition 1.4.2. The dot product has the following algebraic properties: 1. It is commutative : âˆ’â†’ v Â· âˆ’â†’ w = âˆ’â†’ w Â· âˆ’â†’ v 2. It distributes over vector sums 11 : âˆ’â†’ u Â· ( âˆ’â†’ v + âˆ’â†’ w ) = âˆ’â†’ u Â· âˆ’â†’ v + âˆ’â†’ u Â· âˆ’â†’ w 3. it respects scalar multiples : ( r âˆ’â†’ v ) Â· âˆ’â†’ w = r ( âˆ’â†’ v Â· âˆ’â†’ w ) = âˆ’â†’ v Â· ( r âˆ’â†’ w ) . Also, the geometric interpretation of the dot product given by Equation ( 1.17 ) yields a number of geometric properties: 10 Also the scalar product , direct product , or inner product 11 In this formula, âˆ’â†’ u is an arbitrary vector, not necessarily of unit length....
View
Full Document
 Fall '08
 ALL
 Calculus, Factoring, Vectors, Dot Product, scalar product, vw, geometric interpretation, useful geometric interpretation, somewhat diï¬€erent notation

Click to edit the document details