Thursday, May 14, 2020

Revisiting Vectors And Solid Geometry (Part 1)

1.1   Basic Vector Notation:

Consider the vector diagram below:


Here, the resultant is A and the components are:    A x   and A y      such that: 

A =   A x   +  A y        where:

A x    =   A x i       and  A y        =   A y   j    

Where  i  and  j  are unit vectors, and:

A x   =   A cos q ,   A y   =   A sin q ,   q  =  tan -1  (A y  / A x )

  
A x   and  A y    are scalar quantities called the components of the vector A.   Again, the magnitude of A can be written:

A =  Ö (A x  +  A y  )


It is also possible to have more complex vector configurations which require more analysis, such as shown in the diagram below:

In the figure above  the task is to identify the components for each of the resultants, A, B and C.  Given the units are provided to scale, e.g. the resultant A has two equal components of 2 units each, this is possible.


Then for resultant A:

A  =   A x i     +  A y   j    

A x   =  - 2,   A y   =   2  ,

A Ö (A x  +  A y  ) =  Ö (-2)  +  (2)  )

A Ö (8)   =    2 Ö 2

 q  =  tan -1  (A/ A x ) =  tan -1  (2   / -2 ) = tan -1  (-1)

q  =  - 45 degrees

Similarly:

B  =   B x i     +  B y   j    

B x   =  - 2,   B  y   =   -3  ,

B =  Ö (B x  +  B y  ) =  Ö (-2)  +  (-3)  )

B =  Ö (13)  

q  =  tan -1  (B y  / B x ) =  tan -1 (-3   / -2 ) = tan -1  (3/2)

q  =  56. 3 degrees

Finally, we write for resultant C:

C =  Ö (C x  +  C y  ) =  Ö (4)  +  (3)  )

C =  Ö (25)   =   5 units

q  =  tan -1  (C y  / C x ) =  tan -1 (3   /4 ) = tan -1  (3/4)

q  =   36.8 degrees

More generally, we can have a situation with several vectors to be added which need not occur in the same plane, e.g.:
So in this case: T = A + B + C


Whence:

Tx   =   A x      + B x    +  C x    

Ty   =   A y      + B y      +  C y    

Tz   =   A z      + B z      +  C z    


Then once the components  of T have been found the magnitude of the vector may be obtained:

T Ö (T x  +  T y  +   T z   )


By definition the components of such a vector are numbers which multiply the unit vectors .  So if the components of T are  T x,   T y,  and   T z  then:

T  =  T x i   + T y   j   +   T z  k

We can also have the situation for which T = A + B,  but for which the two vectors, A and B, are not necessarily in the same plane. Hence each vector can be described in 3 dimensions using unit vectors. Then one can write:

T  =  (A x i   + A y   j   +   A z  k ) + (B x i   + B y   j   +   B z  k )

And since vector addition is commutative, the preceding can also be written:

T  =  (A x   + B x ) i   + (A y   + B y ) +  (A z   + B z ) k  

We note the length of any vector in 3 dimensions, say referenced along the diagonal of a three- dimensional box, say:


T  =  a  i   + b j   +   c k

Is easily determined by applying the Pythagorean theorem twice, once to the diagonal of a face the rectangular box, then to the overall diagonal, e.g.



In summary:


i   + b j   +   c k Ö (a   +  b  +   c   )

Direction cosines:

If  the diagonal  vector  T  =  a  i   + b j   +   c k    makes angles a, b  and g  respectively, i.e. with the x-, y- and z- axes, then: cos a  ,  cos b    and cos g  are called the direction cosines, where:

cos a  aÖ (a   +  b  +   c   )

cos b    bÖ (a   +  b  +   c   )

cos g   bÖ (a   +  b  +   c   )

It can also be shown that:


cos a  +    cos b    + cos g  =     1


Problems:

1)      For two vectors A = 3i – 4j and B = -ij, find the magnitude and direction of:  A + B,   B A.

2)  Using the method of components, find the vector sum of the two vectors A and B if A makes an angle of 45 degrees with the x-axis and has a length of 6 units, and vector B makes an angle f 135 degrees with the x-axis and has a length of 8 units.

3) For the 3D rectangular box shown earlier, for which: T  =  a    + b j   +   k, assuming  sides a = 6,    b = 3, and c = 2  :

a)Find the direction cosines.


b) Show that:  cos a  +    cos b    + cos g  =     1

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