Vectors

Chapter 9: Vectors

Definition of a Vector

• Vector: A vector in the plane is defined as a line segment with an assigned direction.

• Notation: Vectors are denoted as AB, where:
    - Point A: Initial point of the vector
    - Point B: Terminal point of the vector

• Magnitude: The length of the line segment AB is referred to as the magnitude or length of the vector and is denoted by $| ext{AB}|$.

• Equality of Vectors: Two vectors are considered equal if they have equal magnitude and the same direction.

Operations on Vectors

Addition of Vectors

• Given vectors u and v, the sum is defined as:
    $u + v = (u_{x} + v_{x}, u_{y} + v_{y})$.

Scalar Multiplication

• Let a be a real number and v be a vector. The scalar multiplication is defined as:
    $av$ is defined as follows:
    - The vector $av$ has magnitude $|a| |v|$.
    - The direction of $av$ is the same as $v$ if a > 0, opposite if a < 0, and is the zero vector if $a = 0$.

Vectors in the Coordinate Plane

• A vector v can be represented as an ordered pair of real numbers:
    $v = (a, b)$ where:
    - a: Horizontal component (x-component) of v
    - b: Vertical component (y-component) of v

• Distinction: It is crucial to distinguish between the vector egin{pmatrix} a \ b \ \ ext{and} \ ext{the point} \ (a, b).

Component Form of Vector

• If vector v is represented in the plane with initial point P($x_1, y_1$) and terminal point Q($x_2, y_2$), then the component form is:
    $v = (x_2 - x_1, y_2 - y_1)$.

• In 3D space, if vector v has initial point P($x_1, y_1, z_1$) and terminal point Q($x_2, y_2, z_2$), then:
    $v = (x_2 - x_1, y_2 - y_1, z_2 - z_1)$.

Examples of Vectors

Example 1

1. a) Find the component form of vector u with initial point (−2, 5) and terminal point (3, 7).

2. b) If vector v = (3, 7) has an initial point (2, 4), determine its terminal point.

3. c) Sketch vector w = (2, 3) with initial points (0, 0), (2, 2), (−2, −1), and (1, 4).

Conditions for Vector Equality

• For vectors u = (a1, b1) and v = (a2, b2), the two vectors are equal if:
    $a_1 = a_2 ext{ and } b_1 = b_2$.

• In 3D, vectors u = (a1, b1, c1) and v = (a2, b2, c2) are equal if:
    $a_1 = a_2, b_1 = b_2, ext{ and } c_1 = c_2$.

Magnitude of a Vector

• The magnitude or length of a vector v = (a, b) is given by:
    $|v| = ext{sqrt}(a^2 + b^2)$.

• In R3, the magnitude is:
    $|v| = ext{sqrt}(a^2 + b^2 + c^2)$.

Example 2

• Find the magnitude of each vector:
    1. a) $u = (2, -3)$
    2. b) $v = (5, 0)$
    3. c) w = egin{pmatrix} rac{3}{5} \ rac{4}{5} \ \ ext{(Find the magnitude.)}

Algebraic Operations on Vectors

• If u = (a1, b1) and v = (a2, b2), then:
    1. $u + v = (a1 + a2, b1 + b2)$
    2. $u - v = (a1 - a2, b1 - b2)$
    3. For scalar c: $cu = (ca1, ca2)$, where $c  R$.

• In R3, the same operations are defined as:
    $u + v = (a1 + a2, b1 + b2, c1 + c2)$
    $u - v = (a1 - a2, b1 - b2, c1 - c2)$
    $cu = (ca1, ca2, ca3)$.

Example 3

1. Given u = (2, -3) and v = (-1, 2), calculate:
     - $u + v$
     - $u - v$
     - $2u - 3v$.

Properties of Vectors

1. Commutative Property: $u + v = v + u$.

2. Associative Property: $u + (v + w) = (u + v) + w$.

3. Identity: $u + 0 = u$, where 0 = (0, 0) is the zero vector.

4. Inverse: $u + (-u) = 0$.

5. Scalar Multiplication Distributes Over Vector Addition:
     - $c(u + v) = cu + cv$
     - $(c + d)u = cu + du$.

6. Associative Property of Scalars:
     - $(cd)u = c(du) + d(cu)$.

7. Properties of Scalar 1 and 0:
     - $1u = u$
     - $0u = 0$
     - $c0 = 0$.

8. Length Property: $|cu| = |c||u|$.

9. Triangle Inequality: |u + v|  \leq |u| + |v|.

Unit Vectors

• Unit Vector: A vector of length 1 is called a unit vector.

• Definition: A unit vector is defined by:
    $u = rac{u}{|u|}$.

• Standard Unit Vectors in R2:
    - $e_1 = i = (1, 0)$
    - $e_2 = j = (0, 1)$

• Standard Unit Vectors in R3:
    - $e_1 = i = (1, 0, 0)$
    - $e_2 = j = (0, 1, 0)$
    - $e_3 = k = (0, 0, 1)$.

Expressions in terms of i and j

• A vector v can be expressed in terms of i and j as follows:
    $v = (a, b) = ai + bj$.

• In R3, the expression expands to:
    $v = (a, b, c) = ai + bj + ck$.

Example 4

1. a) Rewrite vector u = (5, −8) in terms of i and j.

2. b) If u = 3i + 2j and v = -i + 6j, write $2u + 5v$ in terms of i and j.

Horizontal and Vertical Components of a Vector

• Let v be a vector with magnitude $|v|$ and direction $heta$.

• The vector can be expressed as:
    $v = (a, b) = |v|cos(θ)i + |v|sin(θ)j$

• Therefore, we can express v as:
    $v = (|v|cos(θ), |v|sin(θ))$.

Example 5

• A dog runs 4 miles in a direction 30° north of east. Assuming the origin is the dog’s starting point and east is the positive x-axis, determine the coordinates of the dog’s location.

Example 6

• Find all vector x which satisfy the vector equation:
    $2x + 5(1, −2, 4) = (5, 12, 17)$.

Example 7

• For each of the vectors given, find a unit vector which points in the same direction as the vector:
    1. a) (1, 2)
    2. b) (2, 3, -5)

Example 8

• A flea hops from point (1, 0) to points (2, 1), (3, 0), (1, -2), and finally (-1, -3). Calculate how far the flea has hopped, and find the vector pointing from the flea’s initial position to its final position.

Dot Product of Vectors

• If u = (a1, b1) and v = (a2, b2) are vectors, then their dot product is defined as:
    u ullet v = a_1a_2 + b_1b_2.

• Note: The dot product results in a scalar (real number), not a vector.

Example 9

• Calculate the dot product for:
    1. a) $u = (3, -2)$ and $v = (4, 5)$.
    2. b) $u = 2i + j$ and $v = 5i - 6j$.

Properties of the Dot Product

1. Commutative: u ullet v = v ullet u.

2. Associative: (au) ullet v = a(u ullet v) = u ullet (av).

3. Distributive: (u + v) ullet w = u ullet w + v ullet w.

4. Length Property: |u|^2 = u ullet u.

Dot Product Theorem

• If $heta$ is the angle between two nonzero vectors u and v, then:

• The relationship is given by:
    u ullet v = |u| |v| cos( heta),

• From this, you can derive:
    cos( heta) = rac{u ullet v}{|u||v|}.

Orthogonal Vectors

• Two nonzero vectors u and v are considered perpendicular (orthogonal) if and only if:
    u ullet v = 0.

Example 10

• Find the angle between the vectors:
    1. $u = (2, 5)$ and $v = (4, -3)$.

Example 11

• Identify all unit vectors that are perpendicular to the vector (2, -3).

Example 12

• An airplane flies from point (1, 1, 3) to point (5, -5, 0.1). The unit distance is one mile. If it takes the plane 4 minutes to make this descent, calculate the descent speed in miles per hour.