# CSCI 2170 Lecture Notes - Lecture 11: Unit Vector, Global Positioning System

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Math 241 Chapter 11 Dr. Justin O. Wyss-Gallifent

§11.1 Cartesian Coordinates in Space

1. Preliminaries: How to plot points in 3-space, the coordinate planes, the ﬁrst octant. Emphasize

how perspective can be confusing at ﬁrst.

2. Distance between points: |P Q|=p(x1−x0)2+ (y1−y0)2+ (z1−z0)2

3. Equation of a circle, a closed disk, a sphere and a closed ball. Pictures of all.

§11.2 Vectors in Space

1. Deﬁnition of a vector as a triple of numbers. The notation ¯aor →

a. We can add and subtract vectors

by adding and subtracting components and we can multiply a scalar by a vector by multiplying

by all the components. Three special vectors are ˆı= (1,0,0), ˆ= (0,1,0) and ˆ

k= (0,0,1).

Then every vector can be written as ¯a=a1ˆı+a2ˆ+a3ˆ

k. Vectors are not necessarily anchored

anywhere though often we anchor them somewhere (the origin, for example) for some reason.

2. Basic properties and associated deﬁnitions:

(a) The zero vector is ¯

0 = 0 ˆı+ 0 ˆ+ 0 ˆ

k.

(b) The length of a vector is ||¯a|| =pa2

1+a2

2+a2

3.

(c) A unit vector has length 1. If a vector ¯ais given we can create a unit vector in the same

direction by doing ¯a/||¯a||.

(d) Two vectors are parallel if they are nonzero multiple of one another. In other words ¯a=c¯

b

with c6= 0.

(e) The vector pointing from P= (a1, a2, a3) to Q= (b1, b2, b3) is →

P Q = (b1−a1) ˆı+(b2−a2) ˆ+

(b3−a3)ˆ

k.

(f) ¯

0 + ¯a= ¯a= ¯a+¯

0

(g) ¯a+¯

b=¯

b+ ¯a

(h) c(¯a+¯

b) = c¯a+c¯

b

(i) 0¯a=¯

0

(j) 1¯a= ¯a

(k) ¯a+ (¯

b+ ¯c) = (¯a+¯

b) + ¯c

3. Geometric interpretation of ¯a+¯

b, of ¯a−¯

band c¯a.

§11.3 The Dot Product

1. Deﬁnition of ¯a·¯

b.

2. Basic properties:

(a) ¯a·¯

b=¯

b·¯a

(b) ¯a·(¯

b+ ¯c) = ¯a·¯

b+ ¯a·¯c

(c) (¯

b+ ¯c)·¯a=¯

b·¯a+ ¯c·¯a

(d) c(¯a·¯

b) = (c¯a)·¯

b= ¯a·(c¯

b)

3. Additional Properties:

(a) If θis the angle between ¯aand ¯

b(anchored at the same point) then ¯a·¯

b=||¯a||||¯

b|| cos θ.

(b) ¯aand ¯

bare perpendicular iﬀ ¯a·¯

b= 0.

(c) ¯a·¯a=||¯a||2.

(d) Deﬁnition of projection of ¯

bonto ¯aand formula Pr¯a¯

b=¯a·¯

b

¯a·¯a¯a.