Geometry of Geometry of RR22 and and RR33

Vectors in RVectors in R22 and R and R33

NOTATION

R The set of real numbers

R2 The set of ordered pairs of real numbers

R3 The set of ordered triples of real numbers

Vector

A vector in R2 (or R3) is a directed line segment from the origin to any point in R2 (or R3)

Vectors inin R R2 2 are represented using ordered pairs

Vectors inin R R3 3 are represented using ordered triples

Notation for Vectors

VectorsVectors in in RR22 (or (or RR33) are denoted using bold faced, lower case, English letters

VectorsVectors in in RR22 (or (or RR33) are written with an arrow above lower case, English letters

Points in in RR22 (or (or RR33) are denoted using upper case English letters

Example 1

u = (u1, u2, u3) represent a vector in R3 from the origin to the point P (u1, u2, u3)

u1, u2, and u3 are the components of the u

Equality of Two Vectors

Two vectors are equal if their corresponding components are equal.

That is, u = (u1, u2, u3) and v = (v1, v2, v3) are equal if and only if u1 = v1, u2 = v2, and u3 = v3

Hence, if u = 0, the zero vector, then u1 = u2 = u3

= 0.

Collinear Vectors

Two vectors are collinear if thy both lie on the same line.

That is, u = (u1, u2, u3) and v = (v1, v2, v3) are collinear if the points U, V, and the Origin are collinear points.

Length of a Vector in R2

The length (norm, magnitude) of v = (v1, v2), denoted by ||v||, is the distance of the point V (v1, v2) from the origin.

22

21 vvv

Length of a Vector in R3

The length (norm, magnitude) of v = (v1, v2, v3) is the distance of the point V (v1, v2, v3) from the origin.

23

22

21 vvvv

Example

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

222 )7()3()4( u

74

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

Let c be a scalar and u a vector in RR22 (or (or RR33). Then the scalar multiple of u by c is the vector the vector obtained by multiplying each component of u by c.

That is, cu = (cu1, cu2) in RR22, , and

cu = (cu1, cu2, cu3) in in RR33

Example

Find cu for u = (-4, 0, 5) and c = 2.

If v = (-1, 1), sketch v, 2v and -2v.

Theorem 1.1.1

Let u be a nonzero vector in RR22 or RR33, 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

Let u and v be nonzero vectors in RR22 or RR33.

Then the sum u + v is obtained by adding the corresponding components.

That is, u + v = (u1 + v1, u2 + v2), in RR22

u + v = (u1 + v1, u2 + v2, u3 + v3), in RR33

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 1.1.2

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

Let u be vector in RR22 or RR33. Then

1. Opposite or Negative of u, denoted by –u, is (-1)(u).

2. The difference u – v is defined as u +(–v).

Theorem 1.1.3

Let u, v and w be vectors in RR22 or RR33, 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.

Let u, v and w be vectors in RR22 or RR33, 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 1.1.4

Let U and V be distinct points in RR22 or RR33. 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.

Standard Basis Vectors in RR22

i = (1, 0)

j = (0, 1)

If (a, b) is a vector in RR22, then

(a, b) = a(1, 0) + b(0, 1) = ai + bj.

Standard Basis Vectors in RR33

i = (1, 0, 0)j = (0, 1, 0)k = (0, 0, 1)

If (a, b, c) is a vector in RR33, then(a, b, c) = ai + bj + ck

Example

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

Homework 1.1