Vectors 1

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Vectors 1
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What is a vector?

• A vector is a mathematical quantity with two
  characteristics
• 1) magnitude how much, size of vector, always
  a non-negative number, length of vector
• 2) Direction orientation in space.
• Can you think of science quantities which are
  vectors in nature?
• Students take notes

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Vectors vs. scalar quantities

• A scalar is a quantity which has only the
  characteristic of magnitude.
• Name some scalars in science...
• Student answers/suggestions

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In art!
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In navigation
New York Harbor
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Prop. 1 Geometrically, a vector is represented
as a ray
V
The terminal or head end
The initial or tail end
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Prop. 2 The length of the ray represents
magnitude and direction of A is the angle the ray
makes with the x axis
y
magnitude
V
?
X
? 0o
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Prop. 3 Two vectors A and B are equal if they
have the same magnitude and direction.
A
B
This property allows us to move vectors around
on our paper/blackboard without changing their
properties.
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Prop. 4 A -B says that vectors A and B are
anti-parallel.They have same size but the
opposite direction.
B
A -B also implies B -A
A
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Prop. 5 Vectors can be added geometrically
Find A B
B
C
A
A
O
B
O
B
Vector C is the sum of A B C A B
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Prop. 6 Vector Addition is CommutativeA B
B A
B
Find A B
A
A
C
A
O
B
O
B
Vector C is the sum of A B C A B B A
This is the parallelogram method learned in
trig.
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Prop. 7 Add vectors head to head
A
A
B
O
B
D
C
C
S
could represent four forces acting upon point 0
-tug-of-war
D
S A B C D
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Prop. 8 Vector subtraction is defined asA - B
A (-B)
-B
A
A
Find A - B
D A - B
B
O
B
O
-B
A
-B
B
Note that the magnitude of D could be larger
than the magnitude of the sum C.
C A B
D A - B
O
B
-B
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Prop. 9 Let A be a vector and k some scalar
number. Then kA is a vector with magnitude kA.
The sign of k dictates the direction of kA.
2A
A
-3A
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A vector A in the x-y plane can be represented by
its perpendicular components
A
y axis
Components AX and AY can be positive,
negative, or zero. The quadrant that vector A
lies in dictates the sign of the components. Compo
nents are scalars.
AY
X axis
AX
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When the magnitude of vector A is given and its
direction specified then its componentscan be
computed easily
y axis
A
AX Acos?
AY
AY Asin?
?
AX
X axis
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AX is negative whileAY is still positive
AX Acos?
A
y axis
AY Asin?
AY
?
The quadrant that A lies in dictates the sign of
the trig functions.
AX
X axis
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Magnitude and direction of a vector can be found
by knowing its components
y axis
A
AY
tan? AY/AX
?
X axis
AX
? tan-1(AY/AX)
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That is all for this slide file....
for this moment...
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