Transcript Scalars and Vectors

Basic Math
Vectors and Scalars
Addition/Subtraction of Vectors
Unit Vectors
Dot Product
Scalars and Vectors (1)
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Scalar – physical quantity that is specified in terms of a single
real number, or magnitude
 Ex. Length, temperature, mass, speed
Vector – physical quantity that is specified by both magnitude
and direction
 Ex. Force, velocity, displacement, acceleration
We represent vectors graphically or quantitatively:
 Graphically: through arrows with the orientation representing the
direction and length representing the magnitude
 Quantitatively: A vector r in the Cartesian plane is an ordered pair
of real numbers that has the form <a, b>. We write
r=<a, b> where a and b are the components of vector v.
Note: Both r and r represent vectors, and will be used
interchangeably.
Scalars and Vectors (2)
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The components a and b are both scalar quantities.
The position vector, or directed line segment from
the origin to point P(a,b), is r=<a, b>.
The magnitude of a vector (length) is found by using
the Pythagorean theorem:
r  r  a, b 
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a 2  b2
Note: When finding the magnitude of a vector fixed
in space, use the distance formula.
Operations with Vectors (1)
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Vector Addition/Subtraction
The sum of two vectors, u=<u1, u2> and
v=<v1, v2> is the vector
u+v =<(u1+v1), (u2+v2)>.
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Ex. If u=<4, 3> and v=<-5, 2>, then u+v=<-1, 5>
Similarly, u-v=<4-(-5), 3-2>=<9, 1>
Operations with Vectors (2)
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Multiplication of a Vector by Scalar
If u=<u1, u2> and c is a real number,
the scalar multiple cu is the vector
cu=<cu1, cu2>.
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Ex. If u=<4, 3> and c=2, then cu=<(2·4), (2·3)>
cu=<8, 6>
Unit Vectors (1)
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A unit vector is a vector of length 1.
They are used to specify a direction.
By convention, we usually use i, j and k to represent
the unit vectors in the x, y and z directions,
respectively (in 3 dimensions).
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i=<1, 0, 0>
j=<0, 1, 0>
k=<0, 0, 1>
points along the positive x-axis
points along the positive y-axis
points along the positive z-axis
Unit vectors for various coordinate systems:
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Cartesian: i, j, and k
Cartesian: we may choose a different set of unit vectors,
e.g. we can rotate i, j, and k
Unit Vectors (2)
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To find a unit vector, u, in an arbitrary direction, for
example, in the direction of vector a, where a=<a1,
a2>, divide the vector by its magnitude (this
process is called normalization).
u
a
a
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
1
a12  a22
a
1
a12  a22
 a1 , a2 
Ex. If a=<3, -4>, then <3/5, -4/5> is a unit vector in the
same direction as a.
Dot Product (1)
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The dot product of two vectors is the sum of the
products of their corresponding components. If
a=<a1, a2> and b=<b1, b2>, then a·b= a1b1+a2b2 .
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Ex. If a=<1,4> and b=<3,8>, then a·b=3+32=35
If θ is the angle between vectors a and b, then
a  b  a b cos
Note: these are just two ways of expressing the dot product
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Note that the dot product of two vectors produces a
scalar. Therefore it is sometimes called a scalar
product.
Dot Product (2)
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Convince yourself of the following:
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a b  a b cos  a cos b  proj(a.on.b ) b
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Conclusion: After you define the direction of an
arbitrary vector in terms of the Cartesian system,
you can find the projection of a different vector onto
the arbitrary direction. By dividing the above
equation by the magnitude of b, you can find the
projection of a in the b direction (and vice versa).
a b
b
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 proj(a.on.b )