Vectors in the Plane

Definition and Representation

Vectors are fundamental mathematical objects characterized by both magnitude and direction. They are commonly represented as directed line segments in the plane, with a tail and a head indicating their orientation.

• Magnitude: The length of the vector.

• Direction: The angle between the vector and the positive x-axis.

• Notation: A vector u is often written as <u1, u2> in component form.

Equal and Parallel Vectors

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

• Parallel Vectors: Vectors u and v are parallel if one is a scalar multiple of the other: for some constant .

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a real number (scalar), affecting its magnitude and possibly its direction.

• If , the direction remains the same.

• If , the direction is reversed.

• The magnitude becomes times the original.

Vector Operations

Addition and Subtraction

Vector addition and subtraction are performed component-wise:

• Addition:

• Subtraction:

Position Vector

A position vector is a vector whose tail is at the origin and whose head is at the point . It is written as .

Component Form and Magnitude

• Component Form:

• Magnitude:

Unit Vector

A unit vector has a magnitude of 1 and points in the direction of a given vector.

• To find a unit vector in the direction of , divide by its magnitude:

Example: Finding Vectors in Component Form

Given points and :

• Magnitude:

• Unit vector:

Vectors in Three Dimensions

3-D Coordinate System

In three dimensions, points are represented as ordered triples . The z-axis is perpendicular to both the x- and y-axes.

• Distance Formula:

• Midpoint Formula:

Vector in 3-D

A vector in space is written as .

• Magnitude:

Standard Unit Vectors in 3-D

• (x-direction)

• (y-direction)

• (z-direction)

Dot Product

Definition and Properties

The dot product (scalar product) of two vectors and is defined as:

• where is the angle between and .

• In component form:

• The dot product is a scalar.

Properties

• Vectors are orthogonal if

Example: Calculating Dot Product and Angle

Given and :

• Magnitude: ,

• Angle:

Conic Sections and Polar Coordinates

The Hyperbola

A hyperbola is the set of all points such that the absolute value of the difference of the distances from two fixed points (the foci) is constant.

• Standard Equation:

• Vertices:

• Foci: where

• Eccentricity:

• Asymptotes:

Conic Sections in Polar Coordinates

Conic sections can be unified and represented in polar coordinates, especially when one focus is at the pole (origin).

• General Polar Equation:

• Eccentricity (): Determines the type of conic:

  • : Ellipse

  • : Parabola

  • : Hyperbola

Example: Polar Equation of an Ellipse

Given , identify the eccentricity and plot points.

• Eccentricity: (ellipse)

• Plot points for various values to sketch the graph.

Applications of Vectors

Physical Quantities

Vectors are used in physics to represent quantities such as velocity and force. For example, the velocity of an airplane relative to the ground can be modeled as the sum of its velocity relative to the air and the wind velocity.

• Resultant velocity:

• Speed: Magnitude of the resultant vector

• Direction: Calculated using the arctangent of the vector components

Geometric Objects in Space

The Sphere and the Ball

• Sphere: The set of all points at a fixed distance from a center .

  • Equation:

• Ball: The set of all points on and inside the sphere.

  • Equation:

Example: Sphere Passing Through Two Points

Given points and , find the equation of the sphere with center at the midpoint of and passing through both points.

• Midpoint:

• Radius:

• Equation:

Summary Table: Vector Operations in Component Form