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
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
(b) The length of a vector is ||¯a|| =pa2
(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¯
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+
0 + ¯a= ¯a= ¯a+¯
b) = c¯a+c¯
(j) 1¯a= ¯a
(k) ¯a+ (¯
b+ ¯c) = (¯a+¯
b) + ¯c
3. Geometric interpretation of ¯a+¯
b, of ¯a−¯
§11.3 The Dot Product
1. Deﬁnition of ¯a·¯
2. Basic properties:
b+ ¯c) = ¯a·¯
b) = (c¯a)·¯
3. Additional Properties:
(a) If θis the angle between ¯aand ¯
b(anchored at the same point) then ¯a·¯
b|| cos θ.
(b) ¯aand ¯
bare perpendicular iﬀ ¯a·¯
(d) Deﬁnition of projection of ¯
bonto ¯aand formula Pr¯a¯