Vector Algebra and Calculus

From the study guide

Course Outline

  1. Vector Definition and Classification
  2. Components and Orthonormal Bases
  3. Vector Addition and Scalar Multiplication
  4. Linear Combinations and Independence
  5. Scalar Product and Vector Norm
  6. Orthogonality and Orthonormality
  7. Vector Product
  8. Mixed and Double Vector Products
  9. Vector Projections and Division
  10. Sine Rule for Triangles
  11. Gradient, Divergence, and Curl

1. Vector Definition and Classification

Key Concepts & Definitions

  • Vector: A directed segment with an origin and endpoint, defined by its origin, direction, sense, and magnitude.
  • Free vector: A specified direction, sense, and magnitude, but an unspecified supporting line and point of application.
  • Sliding vector: A point of application that is not fixed.
  • Bound vector: All its elements determined.
  • Unit vector: A vector whose magnitude equals 1.

Essential Points

  • The vector space considered is the usual three-dimensional Euclidean space.

2. Components and Orthonormal Bases

Key Concepts & Definitions

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Practice quiz preview

1. Which set of attributes defines a vector represented by a directed segment?

2. A vector is known to have a direction, sense, and magnitude, but neither its supporting line nor its point of application is specified. How is it classified?

3. A coordinate basis has pairwise perpendicular basis vectors, but one of those vectors has magnitude 2. Which description applies?

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Flashcard preview

What is a vector?

A directed segment defined by its origin, direction, sense, and magnitude.

What makes a basis orthonormal?

Its basis vectors are mutually perpendicular and each has magnitude 1.

What does the dot product of basis vectors e⃗i\vec{e}_i and e⃗j\vec{e}_j equal when i=ji=j?

1.

What is the dimension of the Euclidean space considered?

Three dimensions.

How are two vectors in R3\mathbb{R}^3 added componentwise?

Their corresponding components are added.

What components result from adding vectors (a1,a2,a3)(a_1,a_2,a_3) and (b1,b2,b3)(b_1,b_2,b_3)?

(a1+b1, a2+b2, a3+b3)(a_1+b_1,\ a_2+b_2,\ a_3+b_3)

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Frequently asked questions

What does the Vector Algebra and Calculus study guide cover?

It covers the core concepts of Vector Algebra and Calculus, organized by topic with the key definitions, explanations and takeaways, so it's easier to learn and remember.

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How many questions are in the Vector Algebra and Calculus quiz?

The practice quiz has 11 multiple-choice questions, each with an explanation. Use it to check what you know and find your gaps.

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How do I study Vector Algebra and Calculus with flashcards?

There are 10 flashcards on Vector Algebra and Calculus, with a question on the front and the answer on the back. Try to recall each answer before you flip, and let spaced repetition bring back the ones you miss.

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