Vector Analysis II

Products Involving Vectors

Introduction

 In this section we will discuss the three different types of products that involve vectors. First we will see what happens when we multiply a scalar and a vector. Secondy, we will learn how to multiply two vectors to obtain a scalar. Finally, we will learn how to multiply two vectors in such a way that another vector is obtained.

Product of a scalar and a vector

 The product of a scalar and a vector is another vector with the same direction as the original vector, but with a different magnitude. If m is a scalar, when a vector A is multiplied by m, their product is
->
mA
where the magnitude of the new vector is |m| times the magnitude of the original vector. If the scalar is negative, the direction of the new vector is exactly the opposite of the original vector.

Scalar Product

 Given two vectors, their scalar product is defined as the product of their magnitudes times the cosine of the angle between them. Analytically:
A*B=|A||B|Cosø
In this type of product, we multiply two vectors in such a way that the result is a scalar (a number), not a vector.

The properties of the scalar product are as follows:

(1) Commutative property
A*B=B*A

(2) Distributive property of the scalar product with respect to addition:
A*(B+C)=A*B + A*C

(3) Multiplication by a scalar:
m(A*B)= (mA)*B= A*(mB)

(4) Product of base vectors
^^´ ^^´ ^^´´ ´
i*i=j*j=k*k=1
^^´ ^^´ ^^´´ ´
i*j=j*k=k*i=0

(5)Given
A=Axi + Ayj +Azk
B=Bxi + Byj + Bzk
it can be shown that
A*B = AxBx + AyBy = AzBz
A*A = |A|² = Ax² + Ay² +Az²
B*B = |B|² = Bx² + By² + Bz²

(6) If the scalar product of two vectors is 0 and both vectors are different than 0, then the vectors are perpendicular.


Vector product

 Given two vectors, their vector product is another vector, denoted by

C = A * B

The magnitude of the new vector is the product of the magnitudes of the vectors that were multiplied times the sine of the angle between them. The direction of the new vector is perpendicular to the plane formed by teh vectors that were multiplied, and pointing in such a way that the three vectors form a right hand system. A right hand system is illustrated by teh figuresgiven below. Placing the fingersof teh right hadn as indicated, the direction of the vector product is indicated by the position of the thumb.

Analytically:

A * B = |A| |B| Sinøu
The properties of the vector product are the following:
  1. The vector product is anticommutative:
    A*B=-B*A
  2. Distributive property of the vector product with respect to the addition
    A*(B*C)=A*B + A*C
  3. Multiplication by a scalar
    m(A*B) = (mA)*B = A*(mB)
  4. Products of base vectors
    i*i = j*j = k*k = 0
    i*j=k; j*k=i; k*i=j
  5. It can be shown that the vector product can be obtained from obtaining a solution to teh following determinant:
    A*B=|i j k|

    |Ax Ay Az|

    |Bx By Bz| It must be noted that the order used ot write the elements of the determinant. It is very important to write the base vectors in teh top row, the first vector of teh product in teh second row, and the second vector of the product in teh last row, because the vector product is anti-commutative.

  6. If the vector product equals 0 adn both vectors are not equal to 0, they are parallel.

    The magnitude fo the vector product is equivalent to the area of teh parallelogram determined by teh two vectors multiplied.


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