Geometry of R 2 and R 3 Vectors in R 2 and R 3

NOTATION RThe set of real numbers R 2 The set of ordered pairs of real numbers R 3 The set of ordered triples of real numbers

Vector A vector in R 2 (or R 3 ) is a directed line segment from the origin to any point in R 2 (or R 3 ) in R 2 Vectors in R 2 are represented using ordered pairs in R 3 Vectors in R 3 are represented using ordered triples

Notation for Vectors Vectors in R 2 (or R 3 Vectors in R 2 (or R 3 ) are denoted using bold faced, lower case, English letters Vectors in R 2 (or R 3 Vectors in R 2 (or R 3 ) are written with an arrow above lower case, English letters in R 2 (or R 3 Points in R 2 (or R 3 ) are denoted using upper case English letters

Example 1 u = (u 1, u 2, u 3 ) represent a vector in R 3 from the origin to the point P (u 1, u 2, u 3 ) u 1, u 2, and u 3 are the components of the u

Equality of Two Vectors Two vectors are equal if their corresponding components are equal. That is, u = (u 1, u 2, u 3 ) and v = (v 1, v 2, v 3 ) are equal if and only if u 1 = v 1, u 2 = v 2, and u 3 = v 3 Hence, if u = 0, the zero vector, then u 1 = u 2 = u 3 = 0.

Collinear Vectors Two vectors are collinear if thy both lie on the same line. That is, u = (u 1, u 2, u 3 ) and v = (v 1, v 2, v 3 ) are collinear if the points U, V, and the Origin are collinear points.

Length of a Vector in R 2 The length (norm, magnitude) of v = (v 1, v 2 ), denoted by ||v||, is the distance of the point V (v 1, v 2 ) from the origin.

Length of a Vector in R 3 The length (norm, magnitude) of v = (v 1, v 2, v 3 ) is the distance of the point V (v 1, v 2, v 3 ) from the origin.

Example Find the length of u = (-4, 3, -7)

Zero Vector and Unit Vector The magnitude of 0 is zero. If a vector has length zero, then it is 0 If a vector has magnitude 1, it is called a unit vector.

Scalar Multiplication R 2 (or R 3 Let c be a scalar and u a vector in R 2 (or R 3 ). Then the scalar multiple of u by c is the vector the vector obtained by multiplying each component of u by c. R 2, That is, cu = (cu 1, cu 2 ) in R 2, and in R 3 cu = (cu 1, cu 2, cu 3 ) in R 3

Example Find cu for u = (-4, 0, 5) and c = 2. If v = (-1, 1), sketch v, 2v and -2v.

Theorem R 2 R 3 Let u be a nonzero vector in R 2 or R 3, and c be any scalar. Then u and cu are collinear, and a) if c > 0, then u and cu have the same direction b) if c < 0, then u and cu have opposite directions c) ||cu|| = |c| ||u||

Example Let u = (-4, 8, -6) a) Find the midpoint of the vector u. b) Find a the unit vector in the direction of u. c) Find a vector in the direction opposite to u that is 1.5 times the length of u.

Vector Addition R 2 R 3 Let u and v be nonzero vectors in R 2 or R 3. Then the sum u + v is obtained by adding the corresponding components. That is, R 2 u + v = (u 1 + v 1, u 2 + v 2 ), in R 2 R 3 u + v = (u 1 + v 1, u 2 + v 2, u 3 + v 3 ), in R 3

Example Find the sum of each pair of vectors 1. u = (2, 1, 0) and v = (-1, 3, 4) 2. u = (1, -2) and v = (-2, 3) Sketch each vector in part (2) and their sum.

Theorem For nonzero vectors u and v the directed line segment from the end point of u to the endpoint of u + v is parallel and equal in length of v.

Proof of Theorem 2: Outline 1) Show that d(u, u+v) = d(0, v). 2) Show that d(v, u+v) = d(0, u). 3) The above two parts proves that the four line segments form a parallelogram. 4) The opposite sides of a parallelogram are parallel and of the same length. (A result from Geometry.) 5) We must also prove that the four vectors u, v, u + v, and 0 are coplanar, which will be done in section 1.2.

Opposite and Vector Subtraction R 2 R 3 Let u be vector in R 2 or R 3. Then 1. Opposite or Negative of u, denoted by –u, is (-1)(u). 2. The difference u – v is defined as u +(–v).

Theorem R 2 R 3 Let u, v and w be vectors in R 2 or R 3, and c and d scalars. Then 1. u + v = v + u 2. (u + v) + w = v + (u + w) 3. u + 0 = u 4. u + (-u) = 0 5. (cd)u = c(du)

Theorem 3 Cont’d. R 2 R 3 Let u, v and w be vectors in R 2 or R 3, and c and d scalars. Then 6. (c + d)u = cu + du 7. c(u + v) = cu + cv 8. 1u = u 9. (-1)u = -u 10. 0u = 0

Equivalent Directed Line Segments Two directed line segments are said to be equivalent if they have the same direction and length.

Theorem R 2 R 3 Let U and V be distinct points in R 2 or R 3. Then the vector v – u is equivalent to the directed line segment from U to V. That is, 1. The line UV is parallel to the vector v – u, and 2. d(u, v) = ||v – u||

Proof of Theorem 4: Outline 1. Show that the sum of u and v – u is v. 2. This proves that the two vectors v – u is parallel and equal in length to the directed line segment from U to V.

Example Is the line determined by (3,1,2) & (4,3,1), parallel to the line determined by (1,3,-3) & (-1,-1,-1)? Outline for the solution: Find unit vectors in the direction of the lines. If they are same or opposite, then the two vectors are parallel.

R 2 Standard Basis Vectors in R 2 i = (1, 0) j = (0, 1) R 2 If (a, b) is a vector in R 2, then (a, b) = a(1, 0) + b(0, 1) = ai + bj.

R 3 Standard Basis Vectors in R 3 i = (1, 0, 0) j = (0, 1, 0) k = (0, 0, 1) R 3 If (a, b, c) is a vector in R 3, then (a, b, c) = ai + bj + ck

Example Express (2, 0, -3) in i, j, k form.

Homework 1.1