{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

dot_product

# dot_product - The Dot Product Thus far we have discussed...

This preview shows pages 1–3. Sign up to view the full content.

T he D ot P roduct Thus far we have discussed the addition, subtraction, and scalar multiplication of vectors. Another operation on vectors is called the dot product . It is important because it can be used to compute the angle between two vectors. There are other properties of the dot product which are covered in a calculus course. We begin by defining the dot product. Dot Product Let 1 2 3 v v v v i j k and 1 2 3 w w w w i j k . The dot product of v and w , denoted by , is a real number given by v w 1 1 2 2 3 3 v w v w v w v w Example 1: Compute if v w 4 3 0 v i j k and 2 2 5   w i j k . Solution: . 4( 2) ( 3)(2) (0)(5) 14     v w As we mentioned above, the dot product can be used to find the angle between two vectors. The relationship is given by the following: Angle Between Two Vectors If v and w are nonzero vectors, then cos v w v w Thus, the angle q between v and w is given by 1 cos v w v w The vectors v and w are perpendicular if and only if . 0 v w To see why the first formula holds, consider the following diagram. For the sake of simplicity, the vectors shown are only in two dimensions. 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document