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Math 241 Chapter 11

Dr
...
Wyss-Gallifent

§11
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Preliminaries: How to plot points in 3-space, the coordinate planes, the first octant
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2
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Equation of a circle, a closed disk, a sphere and a closed ball
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§11
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Definition of a vector as a triple of numbers
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We can add and subtract vectors
¯
by adding and subtracting components and we can multiply a scalar by a vector by multiplying
ˆ
by all the components
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ı
ˆ
ˆ Vectors are not necessarily anchored
Then every vector can be written as a = a1 ˆ + a2  + a3 k
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2
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0
ı
ˆ
(b) The length of a vector is ||¯|| = a2 + a2 + a2
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If a vector a is given we can create a unit vector in the same
¯
direction by doing a/||¯||
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In other words a = c¯
¯
b
with c = 0
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(f) ¯ + a = a = a + ¯
0 ¯ ¯ ¯ 0
(g) a + ¯ = ¯ + a
¯ b b ¯
(h) c(¯ + ¯ = c¯ + c¯
a b)
a
b
(i) 0¯ = ¯
a 0
(j) 1¯ = a
a ¯
(k) a + (¯ + c) = (¯ + ¯ + c
¯
b ¯
a b) ¯
3
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¯ b, ¯ b
a
§11
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Definition of a · ¯
¯ b
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Basic properties:
(a)
(b)
(c)
(d)

a·¯= ¯·a
¯ b b ¯
a · (¯ + c) = a · ¯ + a · c
¯ b ¯
¯ b ¯ ¯
¯ + c) · a = ¯ · a + c · a
(b ¯ ¯ b ¯ ¯ ¯
c(¯ · ¯ = (c¯) · ¯ = a · (c¯
a b)
a b ¯
b)

3
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¯
b
¯ b
a b||
¯ are perpendicular iff a · ¯ = 0
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¯ ¯
a
a·¯
¯b
(d) Definition of projection of ¯ onto a and formula Pra¯ = a·¯ a
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4 The Cross Product
1
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2
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Additional properties:
(a) a × ¯ is perpendicular to both a and ¯ This is extremely useful
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¯ = ||¯||||¯ sin θ
(b) ||¯ × b||
a
a b||
(c) a and ¯ are parallel iff a × ¯ = ¯ but this is not a particularly good way to check
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5 Lines in Space
1
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If
a ˆ + ¯  + c k is parallel to the line and if the line contains the point P = (x0 , y0 , z0 ) then:
¯ ı b ˆ ¯ˆ
2
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Each t gives a point on the line
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Vector Equation: r = (x0 + at) ˆ+ (y0 + bt)  + (z0 + ct) k
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This is far from unique since on any given line there are many
points and many vectors pointing along the line
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Symmetric Equations: Solve for t in each of the parametric equations and set them all equal
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If two do not then leave those alone and don’t even
write the third because that variable could be anything
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If a line has point P and vector L then the distance from another point Q to the line is

¯
||L×P Q||

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6 Planes in Space
ˆ
¯
1
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A point Q = (x, y, z) is on the plane iff the vector from P to Q is
→
ˆ
¯
perpendicular to N , meaning (a ˆ+ b  + c k) · P Q = 0 which is a(x − x0 ) + b(y − y0 ) + c(z − z0 ) = 0
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¯
2
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¯
||N ||

3
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• Those like z = 0 or x = 2 or y = −3, parallel to the coordinate planes
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