Overview
On this page, we discuss different ways to “combine” vectors to form new ones, and a first method of describing lines and planes in 3-dimensional space.
Important
The basic and advanced learning objectives listed below are meant to give you an idea of the material you should learn about this section. These are mainly intended to be used in a course which uses an Active Learning approach, where students are required to “read ahead” before each class - but can equally be used in a more traditional course setting.
Unless your teacher gives you specific instructions, it is up to you to decide how much of the listed resources you need to read or watch - you probably do not need to go through all of it. You might also want to look at the General Study Tips & Tricks page for some recommendations on how to effectively study with a math textbook and videos.
Basic learning objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
- Given a collection of vectors, form a new vector as a linear combination of the given ones.
- Verify if a given vector can be expressed as a linear combination of a set of other vectors.
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
- Express the equations of lines and planes in \(\mathbb{R}^2\), \(\mathbb{R}^3\), and \(\mathbb{R}^n\) using either vector or parametric equations.
- Express the equations of a line containing two given points in \(\mathbb{R}^2\) or \(\mathbb{R}^3\) using either vector or parametric equations.
To prepare for class
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Watch this video (by 3Blue1Brown) which explains linear combinations of vectors, and the collection of all possible vectors which can be created that way (called the “span”):
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Watch this video which explains the standard unit vectors in \(\mathbb{R}^2\) and how all other vectors in \(\mathbb{R}^2\) are in fact linear combinations of these two:
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Watch these two videos explaining how to find the parametric equation(s) of a line in 3-space, and a detailed example:
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Watch the beginning of this video explaining how to find the parametric equation(s) of a plane in 3-space, and a detailed example (ignore everything after 4:20 for now, as it shows an alternative way to describe the same plane, using a “normal equation”):
After class
- Play around with the following interactive applets (created by Gabriel Indurskis) on Desmos (make sure to read the instructions):
- Parametrization of a line in \(\mathbb{R}^3\): https://www.desmos.com/3d/uqr7qigep4
- Parametrization of a plane in \(\mathbb{R}^3\): https://www.desmos.com/3d/llepnyv69l
