Vectors in The Plane, Space - Revision history

Lila: /* Resources */

2021-11-02T16:02:48Z

Resources

Lila at 21:04, 18 October 2021

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Lila at 19:34, 1 October 2021

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Lila: /* Exercises */

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Exercises

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Exercises

Lila at 19:20, 1 October 2021

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Khanh: Created page with "* [https://math.libretexts.org/Courses/Monroe_Community_College/MTH_212_Calculus_III/Chapter_11%3A_Vectors_and_the_Geometry_of_Space/11.1%3A_Vectors_in_the_Plane Vectors in th..."

2021-09-25T21:41:33Z

Created page with "* [https://math.libretexts.org/Courses/Monroe_Community_College/MTH_212_Calculus_III/Chapter_11%3A_Vectors_and_the_Geometry_of_Space/11.1%3A_Vectors_in_the_Plane Vectors in th..."

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* [https://math.libretexts.org/Courses/Monroe_Community_College/MTH_212_Calculus_III/Chapter_11%3A_Vectors_and_the_Geometry_of_Space/11.1%3A_Vectors_in_the_Plane Vectors in the Plane], Mathematics LibreTexts

← Older revision		Revision as of 16:02, 2 November 2021
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	==Resources==		==Resources==
	* [https://math.libretexts.org/Courses/Monroe_Community_College/MTH_212_Calculus_III/Chapter_11%3A_Vectors_and_the_Geometry_of_Space/11.1%3A_Vectors_in_the_Plane Vectors in the Plane], Mathematics LibreTexts		* [https://math.libretexts.org/Courses/Monroe_Community_College/MTH_212_Calculus_III/Chapter_11%3A_Vectors_and_the_Geometry_of_Space/11.1%3A_Vectors_in_the_Plane Vectors in the Plane], Mathematics LibreTexts
		+
		+	==Licensing==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikibooks.org/wiki/Linear_Algebra/Vectors_in_Space Vectors in Space, Wikibooks: Linear Algebra] under a CC BY-SA license

← Older revision		Revision as of 21:04, 18 October 2021
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		+	''See also: [[Vectors]]''
		+
	"Higher-dimensional geometry" sounds exotic. It is exotic— interesting and eye-opening. But it isn't distant or unreachable.		"Higher-dimensional geometry" sounds exotic. It is exotic— interesting and eye-opening. But it isn't distant or unreachable.

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 </math>
-{{anchor|vector operations}}In the prior section we defined vectors and vector operations with an algebraic motivation;
+===Vector Operations===
 :<math>
@@ Line 171: / Line 172: @@
 [[File:Linalg plane with basis.png|center]]
-{{anchor|linear surface}}Generalizing from lines and planes, we define a '''<math> k </math>-dimensional linear surface''' (or '''<math> k </math>-flat''') in <math>\mathbb{R}^n</math> to be <math>\{\vec{p}+t_1\vec{v}_1+t_2\vec{v}_2+\cdots+t_k\vec{v}_k \,\big|\, t_1,\ldots ,t_k\in\mathbb{R}\}</math> where <math> \vec{v}_1,\ldots,\vec{v}_k\in\mathbb{R}^n </math>. For example, in <math>\mathbb{R}^4</math>,
+===Linear Surface===
 :<math>

@@ Line 225: / Line 225: @@
 We finish this subsection by restating our conclusions from the first section in geometric terms. First, the solution set of a linear system with <math> n </math> unknowns is a linear surface in <math> \mathbb{R}^n </math>. Specifically, it is a <math> k </math>-dimensional linear surface, where <math> k </math> is the number of free variables in an echelon form version of the system. Second, the solution set of a homogeneous linear system is a linear surface passing through the origin. Finally, we can view the general solution set of any linear system as being the solution set of its associated homogeneous system offset from the origin by a vector, namely by any particular solution.
-== Exercises ==
+== Practice Problems ==
-{{Linear Algebra/Book 2/Recommended}}
+[https://en.wikibooks.org/wiki/Linear_Algebra/Vectors_in_Space/Solutions Vectors in Space (With Solutions)]
 ==Resources==
 * [https://math.libretexts.org/Courses/Monroe_Community_College/MTH_212_Calculus_III/Chapter_11%3A_Vectors_and_the_Geometry_of_Space/11.1%3A_Vectors_in_the_Plane Vectors in the Plane], Mathematics LibreTexts

@@ Line 247: / Line 247: @@
 from <math> (5,1,4) </math> to <math> (6,0,7) </math>
 </ol>
-}}
-{{Linear Algebra/Book 2/Recommended}}
+Problem 3: Does <math> (1,0,2,1) </math> lie on the line through <math> (-2,1,1,0) </math> and <math> (5,10,-1,4) </math>?
-{{TextBox|1=
-;Problem 3:
+Problem 4:
 <ol type=1 start=1>
 <li> Describe the plane through <math> (1,1,5,-1) </math>,
@@ Line 262: / Line 256: @@
 <li> Is the origin in that plane?
 </ol>
-}}
-{{TextBox|1=
+Problem 5: Describe the plane that contains this point and line.
 :<math>
@@ Line 275: / Line 267: @@
 </math>
-}}
+Problem 6: Intersect these planes.
 :<math>
@@ Line 292: / Line 280: @@
 </math>
-}}
+Problem 7: Intersect each pair, if possible.
-{{Linear Algebra/Book 2/Recommended}}
 <ol type=1 start=1>
 <li> <math> \{\begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}+t\begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}
@@ Line 308: / Line 293: @@
 \,\big|\, s,w\in\mathbb{R}\} </math>
 </ol>
-}}
-{{TextBox|1=
-;Problem 8:
+Problem 8: When a plane does not pass through the origin, performing operations on vectors whose bodies lie in it is more complicated than when the plane passes through the origin. Consider the picture in this subsection of the plane
 :<math>
@@ Line 331: / Line 315: @@
 The endpoint of the sum, on the diagonal, is not <math>(3,1,1)</math>; what is it?
 </ol>
 ==Resources==
 * [https://math.libretexts.org/Courses/Monroe_Community_College/MTH_212_Calculus_III/Chapter_11%3A_Vectors_and_the_Geometry_of_Space/11.1%3A_Vectors_in_the_Plane Vectors in the Plane], Mathematics LibreTexts