The dot product

If the dot product equals zero, the vectors are perpendicular.  If the vectors are multiples of each other, they are parallel.  Formula for dot product:  u · v = · = x1x2 + y1y2 You  use this formula to find the angle between two vectors: cosƟ = u · v/|u||v|   Properties of the dot product 1. Commutative  u · v = v · u u · u ≠ u^2 2. Squared  u · u = |u|^2 3. k(u · v) = ku · v  doesn't distribute 4. u(v + w) = u  · v + u · w Example 1- V=2i+4j   W=i+5j v · w  =  (2)(1) + (4)(5)  =  22 Example 2- Find the angle between  v=  2i + 3j + k and w  =  4i + j + 2k. ||v||=√4+9+1=√14 ||w||=√16+1+4=√21 v . w  =  8 + 3 + 2 = 13 Ɵ=cosƟ^-1((13)/(√14)(√21))