Physics 201 Experiment #1 Vector Algebra
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Objective
In depth study of the algebra of physical quantities described by Vectors. Vector addition, scalar
multiplication of a vector, the scalar product and the vector product will be introduced. Geometric and
algebraic representations of vectors will be studied
Equipment
ruler
protractor
graph paper
calculator
Introduction
Physical quantities that require a specification of both magnitude and direction are represented by
vector quantities. Physical quantities that require a specification of a magnitude only are represented
by scalars.
Vectors may be represented by a geometrical object (a directed line segment ) or equivalently as an
algebraic object with respect to a coordinate basis.
Addition of vectors in the geometric view is accomplished by the tail to head polygon method.
Addition of vectors in the algebraic view is accomplished by the addition of like components.
The algebra of vector addition with scalar multiplication forms the algebraic structure known as a
RING. Vector addition is an ABELIAN GROUP and when scalar multiplication is included our
algebra forms a RING. A group is set of objects with a binary operation defined with the following 4
properties:
1. Closure: For any two vectors, the sum is also a vector
2. Associativity: For any three vectors, A+(B+C) = (A+B)+C
3. Identity element: There exists a vector, 0, such that for any vector A 0+A=A+0=A
4. Invertability: For every vector A, there exists a vector "-A" such that A+(-A) = (-A) +A = 0
If the binary operation is commutative for every element then the group is Abelian
5. Commutativity: For any two vectors A+B = B+A
Scalar multiplication of a vector is a second binary operation that is associative, is distributive with
respect to vector addition and there exist an Identity element.
6. Associativity: For any two scalars and a vector a*(bA) = (ab)*A=b*(aA)*(ba)*A
7. Distributivity: For any scalar "a" and vectors "A" and "B",
a*(A+B)=aA+aB =(A+B)*a=Aa + Ba
8. Identity element: There exists a scalar, 1, such that for any vector A 1A=A1=A
