Math 1C: Calculus III
Part 1: Multidimensional Differential Calculus
Lesson 2: Vectors in 3D
©Jeffrey A. Anderson

Lesson 2 Key Terms List: Briggs Section 11.2
□ 3-Dimensional Cartesian Coordinate System: (ℝ^3) (p. 771)
□ Coordinate Axes (p. 771)
□ Right-handed coordinate system (771)
□ Coordinate Planes (p. 771)
	□ x-plane in (ℝ^3)(p. 771)
	□ xz-plane in (ℝ^3)(p. 771)
	□ yz-plane in (ℝ^3)(p. 771)
□ Equations of Simple Planes in (ℝ^3)(p. 772)
□ Distance Formula in xyz-space (p. 773)
□ Equations of Surfaces in (ℝ^3) (p. 635)
	□ Standard Equation for a Sphere in (ℝ^3)(p. 773)
	□ Standard Equation for a Cylinder in (ℝ^3)(in Jeff’s Lesson Notes)
□ Vectors in (ℝ^3)(p. 774)
□ Vector Operations in (ℝ^3)(p. 775)
□ Magnitude of a Vector in (ℝ^3)(p. 776)

. Section 9.1: Definitions and Visualizations

Example 1: Vectors in 3D

Points in 3D

p = {2,-3,1};
o= {0,0,0};
Show[
	Graphics3D[
		{PointSize[.02],Point[{o,p}]},Axes->True
	],(*end: Graphics3D command*)
	ParametricPlot3D[{{t,0,0}, {0,t,0}, {0,0,t}},{t,-5,5}],
	TicksStyle->Directive[FontSize-> 12],
	AxesLabel->{x,y,z},
	PlotRange->{{-5,5},{-5,5},{-5,5}}
](*end: Show command*)

Vectors in 3D

p = {2,-4,2};
q= {0,0,0};
Show[
	Graphics3D[{PointSize[.02],Point[{o,p}]},Axes->True],
	Graphics3D[{Thickness[0.01],Arrow[{o,p}]},Axes->True],
	ParametricPlot3D[{{t,0,0}, {0,t,0}, {0,0,t}},{t,-5,5}],
	TicksStyle->Directive[FontSize-> 12],
	AxesLabel->{x,y,z},
	PlotRange->{{-5,5},{-5,5},{-5,5}}
](*end: Show command*)

Example 2: Planes in 3D

Visualizing Simple Planes in 3D

ContourPlot3D[
	{z==0, z == 3},
	{x,-4,4},
	{y,-4,4},
	{z,-4,4},
	AxesLabel->{x,y,z}
](*end: ContourPlot3D command*)

Example 3: Spheres in 3D

Visualization of Specific Equation for a Sphere of Radius One

ContourPlot3D[
	(((x-0))^2)+(((y-0))^2)+(((z-0))^2)==1,
	{x,-4,4},
	{y,-4,4},
	{z,-4,4},
	AxesLabel->{x,y,z}
](*end: ContourPlot3D command*)

Example 4: Cylinders in 3D

ContourPlot3D[
	(x-0)^2+(y-0)^2==4,
	{x,-5,5},
	{y,-6,6},
	{z,-2,2}
](*end: ContourPlot3D command*)