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  ²  +  A _y ²  )

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  ²  +  A _y ²  ) =  Ö (-2)  ²  +  (2) ²  )

‖ A ‖ =  Ö (8)   =    2 Ö 2

 q  =  tan ^-1  (A _y  / 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  ²  +  B _y ²  ) =  Ö (-2)  ²  +  (-3) ²  )

‖ 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  ²  +  C _y ²  ) =  Ö (4)  ²  +  (3) ²  )

‖ 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:

T_x   =   A _x      + B _x    +  C _x    

T_y   =   A _y      + B _y      +  C _y    

T_z   =   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  ²  +  T _y ²  +   T _z ²   )

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 ) j  +  (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:

‖ a  i   + b j   +   c k ‖ =  Ö (a  ²  +  b ²  +   c  ²  )

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  =  a /  Ö (a  ²  +  b ²  +   c  ²  )

cos b    =  b /  Ö (a  ²  +  b ²  +   c  ²  )

cos g   =  b /  Ö (a  ²  +  b ²  +   c  ²  )

It can also be shown that:

cos ²  a  +    cos ²  b    + cos ²  g  =     1

Problems:

1)      For two vectors A = 3i – 4j and B = -i – j, 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  i   + b j   +   c k, assuming  sides a = 6,    b = 3, and c = 2  :

a)Find the direction cosines.

b) Show that:  cos ²  a  +    cos ²  b    + cos ²  g  =     1