Basic Math
Vectors and Scalars
Addition/Subtraction of Vectors
Unit Vectors
Dot Product
Scalars and Vectors (1)
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 . We write
r= 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)
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=.
The magnitude of a vector (length) is found by using
the Pythagorean theorem:
r r a, b a 2 b2
Note: When finding the magnitude of a vector fixed
in space, use the distance formula.
Operations with Vectors (1)
Vector Addition/Subtraction
The sum of two vectors, u= and
v= is the vector
u+v =.
Ex. If u= and v=, then u+v=
Similarly, u-v==
Operations with Vectors (2)
Multiplication of a Vector by Scalar
If u= and c is a real number,
the scalar multiple cu is the vector
cu=.
Ex. If u= and c=2, then cu=
cu=
Unit Vectors (1)
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).
i= points along the positive x-axis
j= points along the positive y-axis
k= points along the positive z-axis
Unit vectors for various coordinate systems:
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)
To find a unit vector, u, in an arbitrary direction,
for example, in the direction of vector a, where
a=, divide the vector by its magnitude (this
process is called normalization).
a 1 1
u a a1 , a 2
a a12 a 2
2
a12 a 2
2
Ex. If a=, then is a unit vector in the
same direction as a.
Dot Product (1)
The dot product of two vectors is the sum of the
products of their corresponding components. If
a= and b=, then a·b= a1b1+a2b2 .
Ex. If a= and b=, 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
Note that the dot product of two vectors produces a
scalar. Therefore it is sometimes called a scalar
product.
Dot Product (2)
Convince yourself of the following:
a b a b cos a cos b proj (a.on.b ) b
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
proj (a.on.b )
b