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Geometry of R 2 and R 3PowerPoint Presentation

Geometry of R 2 and R 3

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### Geometry of R2 and R3

Vectors in R2 and R3

NOTATION

- R The set of real numbers
- R2The set of ordered pairs of real numbers
- R3The 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 in R2 are represented using ordered pairs
- Vectors in R3 are represented using ordered triples

Notation for Vectors

- Vectors in R2 (or R3) are denoted using bold faced, lower case, English letters
- Vectors in R2 (or R3) are written with an arrow above lower case, English letters
- Points in R2 (or R3) are denoted using upper case English letters

Example 1

- u = (u1, u2, u3) represent a vector in R3from 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.

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.

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

Let c be a scalar and u a vector in R2 (or R3). 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 R2, and

cu = (cu1, cu2, cu3) in R3

Theorem 1.1.1

Let u be a nonzero vector in R2 or R3, and c be any scalar. Then u and cu are collinear, and

- if c > 0, then u and cu have the same direction
- if c < 0, then u and cu have opposite directions
- ||cu|| = |c| ||u||

Example

Let u = (-4, 8, -6)

- Find the midpoint of the vector u.
- Find a the unit vector in the direction of u.
- 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 R2 or R3.Then the sum u + v is obtained by adding the corresponding components.

That is,

- u + v = (u1 + v1, u2 + v2), in R2
- u + v = (u1 + v1, u2 + v2, u3 + v3), in R3

Example

Find the sum of each pair of vectors

- u = (2, 1, 0) and v = (-1, 3, 4)
- 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

- Show that d(u, u+v) = d(0, v).
- Show that d(v, u+v) = d(0, u).
- The above two parts proves that the four line segments form a parallelogram.
- The opposite sides of a parallelogram are parallel and of the same length. (A result from Geometry.)
- 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 R2 or R3. Then

- Opposite or Negative of u, denoted by –u, is (-1)(u).
- The difference u – v is defined as u +(–v).

Theorem 1.1.3

Let u, v and w be vectors in R2 or R3, and c and d scalars. Then

- u + v = v + u
- (u + v) + w = v + (u + w)
- u + 0 = u
- u + (-u) = 0
- (cd)u = c(du)

Theorem 3 Cont’d.

Let u, v and w be vectors in R2 or R3, and c and d scalars. Then

- (c + d)u = cu + du
- c(u + v) = cu + cv
- 1u = u
- (-1)u = -u
- 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 R2 or R3. Then the vector v – u is equivalent to the directed line segment from U to V. That is,

- The line UV is parallel to the vector v – u, and
- d(u, v) = ||v – u||

Proof of Theorem 4: Outline

- Show that the sum of u and v – u is v.
- 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 R2

i = (1, 0)

j = (0, 1)

If (a, b) is a vector in R2, then

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

Standard Basis Vectors in R3

i = (1, 0, 0)

j = (0, 1, 0)

k = (0, 0, 1)

If (a, b, c) is a vector in R3, then

(a, b, c) = ai + bj + ck

Example

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

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