Geometry of R2 and R3 Dot and Cross Products

Dot Product in R2 Let u = (u1, u2) and v = (v1, v2) then the dot product or scalar product, denoted by u.v, is defined as u.v = u1v1 + u2v2

Dot Product in R3 Let u = (u1, u2, u3) and v = (v1, v2, v3) then the dot product or scalar product, denoted by u.v, is defined as u.v = u1v1 + u2v2 + u3v3

Example 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)

Theorem 1.2.1 Let u and v be vectors in R2 or R3, and let c be a scalar. Then u.v = v.u c(u.v) = (cu).v = u. (cv) u.(v + w) = u.v + u.w u.0 = 0 u.u = ||u||2 Prove c and e in class.

Theorem 1.2.2 Let u and v be vectors in R2 or R3, and let  be the angle they form. Then u.v = ||v|| ||u|| cos If u and v are nonzero vectors, then Proof: 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.

Example Find the angle between each pair of vectors. u = (-1, 2, 3); v = (2, 0, 4) u = (1, 0, 1); v = (-1, -1, 0)

Orthogonal Vectors Two vectors u and v in R2 or R3 are orthogonal if u.v = 0. Orthogonal, Normal, and Perpendicular, all mean the same.

Theorem 1.2.3 Let u and v be nonzero vectors in R2 or R3 and let  be the angle they form. Then  is An acute angle if u.v > 0 A right angle if u.v = 0 An obtuse angle if u.v < 0

Cross 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)

Cross 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:

Example Find the cross product of the following vectors u = (-1, 1, 0); v = (2, 3, -1)

Theorem 1.2.4 The vector uxv is orthogonal to both u and v.

Theorem 1.2.4 Let u, v, and w be vectors in R3, and let c be a scalar. Then u 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 = 0 u x u = 0 ||u x v|| = ||u|| ||v|| sin  = (||u|| ||v|| – ||u.v||2) Do parts c and g.

Cross Product: Area Let 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 

Example Find the area of the parallelogram determined by the vectors u = (-1, 1, 0) and v = (2, 3, -1).

Homework 1.2