Vectors and Geometry in Calculus

Introduction

This study guide covers fundamental concepts of vectors and their applications in geometry, as commonly encountered in college-level Calculus courses. Topics include vector operations, vector components, magnitude, position vectors, equations of spheres, and geometric interpretations in three-dimensional space.

Vector Basics

Vector Representation and Operations

Vectors are quantities that have both magnitude and direction. In two or three dimensions, vectors are often written in component form as ai + bj (2D) or ai + bj + ck (3D), where i, j, and k are unit vectors along the x, y, and z axes, respectively.

• Vector Addition: To add vectors, add their corresponding components.

• Scalar Multiplication: Multiply each component by the scalar.

• Example: If and , then .

Expressing Vectors from Two Points

The vector from point A to point B, denoted , is found by subtracting the coordinates of A from those of B.

• Formula: If and , then .

• Example: If and , then .

Vector Magnitude and Distance

Magnitude of a Vector

The magnitude (or length) of a vector is given by:

• Example: For and , .

• Example: For , .

Distance Between Two Points

The distance between points and is:

• Example: , : .

Vector Decomposition and Linear Combinations

Expressing a Vector as a Linear Combination

Any vector can be written as a linear combination of two non-parallel vectors and : , where and are scalars.

• Set up a system of equations by equating components and solve for and .

• Example: If , , , solve to find , .

Applications of Vectors

Resolving Forces into Components

A force vector can be resolved into its horizontal (i) and vertical (j) components using trigonometry.

• Formulas: For a force at angle with the horizontal:

  • Horizontal:

  • Vertical:

• Example: lb, :

  • So,

Equilibrium and Tension in Cables

When an object is suspended by two cables, the tensions can be found by resolving forces and applying equilibrium conditions.

• Sum of vertical components equals the weight; sum of horizontal components equals zero.

• Example: For a 50 lb speaker suspended by cables at and :

  • Set up equations: (vertical), (horizontal)

  • Solve for and (see table below for sample values).

Position Vectors and 3D Geometry

Position Vector

The position vector from point to in 3D is .

• Example: , : .

Equations of Spheres

The equation of a sphere with center and radius is:

• To find the center and radius from a general equation, complete the square for each variable.

• Example: Center , radius $5$:

  • Equation:

• Example: Given , complete the square to find center and radius $10$.

Geometric Descriptions in 3D

Inequalities involving describe regions in space:

• describes all points outside the sphere of radius 1 centered at the origin.

Summary Table: Key Vector Formulas

Additional info:

• These topics are foundational for multivariable calculus and physics applications.

• Understanding vector operations is essential for later topics such as dot product, cross product, and vector-valued functions.