2Dot Product in R2Let u = (u1, u2) and v = (v1, v2) then the dot product or scalar product, denoted by u.v, is defined asu.v = u1v1 + u2v2
3Dot Product in R3Let u = (u1, u2, u3) and v = (v1, v2, v3) then the dot product or scalar product, denoted by u.v, is defined asu.v = u1v1 + u2v2 + u3v3
4Example Find the dot product of each pair of vectors u = (-3, 2, -1); v = (-4, -3, 0)u = (-4, 0, -2); v = (-3, -7, 6)u = (-6, 3); v = (5, -8)
5Theorem 1.2.1Let u and v be vectors in R2 or R3, and let c be a scalar. Thenu.v = v.uc(u.v) = (cu).v = u. (cv)u.(v + w) = u.v + u.wu.0 = 0u.u = ||u||2Prove c and e in class.
6Theorem 1.2.2Let u and v be vectors in R2 or R3, and let be the angle they form. Thenu.v = ||v|| ||u|| cosIf u and v are nonzero vectors, thenProof: start with the law of cosines. Convert the sides in terms of the norm. Also expand the norm of v-u and set it equal to the previous.
7Example Find the angle between each pair of vectors. u = (-1, 2, 3); v = (2, 0, 4)u = (1, 0, 1); v = (-1, -1, 0)
8Orthogonal VectorsTwo vectors u and v in R2 or R3 are orthogonal if u.v = 0.Orthogonal, Normal, and Perpendicular, all mean the same.
9Theorem 1.2.3Let u and v be nonzero vectors in R2 or R3 and let be the angle they form. Then isAn acute angle if u.v > 0A right angle if u.v = 0An obtuse angle if u.v < 0
10Cross Product (Only in R3 ) Let u = (u1, u2, u3) and v = (v1, v2, v3) be nonzero vectors in R3. Then the cross product, denoted by u x v, is the vector(u2v3 – u3v2, u3v1 – u1v3, u1v2 – u2v1)
11Cross Product (Convenient notation ) Let u = (u1, u2, u3) and v = (v1, v2, v3) be nonzero vectors in R3. Then u x v, is the vector obtained by evaluating the determinant:
12Example Find the cross product of the following vectors u = (-1, 1, 0); v = (2, 3, -1)
13Theorem 1.2.4The vector uxv is orthogonal to both u and v.
14Theorem 1.2.4Let u, v, and w be vectors in R3, and let c be a scalar. Thenu x v = –(v x u)u x (v + w) = (u x v) + (u x w)(u + v) x w = (u x w) + (v x w)c(u x v ) = (cu) x v = u x (cv)u x 0 = 0 x u = 0u x u = 0||u x v|| = ||u|| ||v|| sin = (||u|| ||v|| – ||u.v||2)Do parts c and g.
15Cross Product: AreaLet u, and v, be vectors in R3, Then the area of the parallelogram determined by u and v is||u x v|| = ||u|| ||v|| sin
16ExampleFind the area of the parallelogram determined by the vectors u = (-1, 1, 0) and v = (2, 3, -1).