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The programmable vector calculator performs vector
operations and makes it possible to create unique solutions to otherwise
difficult problems.
Many geometrical problems in 2d or 3d space are most easily
solved by vector methods. Making each point a vector variable some very elegant
solutions are possible.
Maths Helper Plus comes with a selection of geometric problem
solutions based on the vector calculator. You don't need to understand vectors
to make use of these solutions.Â Â
 Program tab 
Type expressions and program instructions into this 'Program
tab'. A full range of vector operations such as scalar and vector products are
supported.
The example below comes with Maths Helper Plus. It finds the
point that divides a line segment internally and externally according to a given
ratio, as well as the midpoint of the line. It works for any number of
dimensions.
The vector variables 'A' and 'B' contain the coordinates of
the two points defining the line segment, while 'C' and 'D' are scalars that
define the ratio of the division. These variables have already been entered in
the 'Vector Editor' tab.
Clicking 'Run' displays these results:
Output
The coordinates of the point X that divide
the line segment AB in the ratio C : D where:
Â A = 1
Â Â Â Â 2
Â Â Â Â 3
Â B = 2
Â Â Â Â 3
Â Â Â Â Â 6
C = 2
D = 3
are given by: (INTERNAL DIVISION)
Â X = [CB + DA] / (C+D)
Â Â = 0.2
Â Â Â Â Â Â Â 0
Â Â Â Â Â 4.2
and: (EXTERNAL DIVISION)
[CB  DA] / (CD) = 7
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â
12
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â
3
The midpoint of the line segment AB is given by:
Â [A + B] / 2 = 0.5
Â Â Â Â Â Â Â Â Â Â Â Â Â Â
0.5
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â
4.5 
Â
 Vector Editor 
Edit the five vector variables in this tab.
 2d or 3d Vector Calculations 
Â Select options in this tab to calculate the basic vector
operations with working steps for vector variables 'A' and 'B':
Here is an example of the output displayed on the text view
for 3d vectors:
Vector calculations
Vector Sum:
Â Â A + B = (xA + xB)i + (yA + yB)j + (zA + zB)k
Â Â Â Â Â Â Â Â = (1 + 2)i
Â Â Â Â Â Â Â Â Â Â + (2 + 3)j
Â Â Â Â Â Â Â Â Â Â Â + (3 +
6)k
Â Â Â Â Â Â Â Â = 1i + 1j + 9k
Â A + B = 9.11043,
Vector Difference:
Â Â A  B = (xA  xB)i + (yA  yB)j + (zA  zB)k
Â Â Â Â Â Â Â Â = (1  2)i
Â Â Â Â Â Â Â Â Â Â + (2  3)j
Â Â Â Â Â Â Â Â Â Â Â + (3 
6)k
Â Â Â Â Â Â Â Â = 3i + 5j + 3k
Â A  B = 6.55744,
Scalar ('dot') product:
Â Â Â Â A·B = (xA × xB) + (yA × yB) + (zA × zB)
Â Â Â Â Â Â Â Â = (1 × 2)
Â Â Â Â Â Â Â Â Â Â + (2 × 3)
Â Â Â Â Â Â Â Â Â Â Â + (3 ×
6)
Â Â Â Â Â Â Â Â = 10
Angle between vectors A and B:
Â Â Â Â A·B = ABcos(T), where 'T' is the
Â Â Â Â angle between 'A' and 'B',
Â Â Â Â Â Â Â So T = acos[ A·B / (AB) ]
Â Â Â Â Â Â Â Â Â Â Â Â =
acos[ 10 / (3.74166 × 7) ]
Â Â Â Â Â Â Â Â Â Â Â Â =
67.5547°
Vector ('cross') product:
Â Â Â Â A×B = [(yA × zB)  (zA × yB)]i
Â Â Â Â Â Â Â Â Â + [(zA × xB)  (xA
× zB)]j
Â Â Â Â Â Â Â Â Â Â Â + [(xA
× yB)  (yA × xB)]k
Â Â Â Â Â Â Â Â = [(2 × 6)  (3 × 3)]i
Â Â Â Â Â Â Â Â Â + [(3 × 2)  (1
× 6)]j
Â Â Â Â Â Â Â Â Â Â Â + [(1
× 3)  (2 × 2)]k
Â Â Â Â Â Â Â Â = 21i + 12j + 1k
Area of parallelogram and perpendicular unit vector:
Â A×B = uABsin(T), where 'u' is a unit vector
Â perpendicular to 'A' and 'B', and 'T' is the angle
Â between 'A' and 'B'.
Â The scalar expression: ABsin(T) = A×B
Â represents the area of the parallelogram
Â defined by 'A' and 'B'.
Â In this case, A×B = 24.2074
Â So 'u' = [1/A×B](A×B)
Â Â Â Â Â Â Â Â = (21/24.2074)i
Â Â Â Â Â Â Â Â Â Â +
(12/24.2074)j
Â Â Â Â Â Â Â Â Â Â Â +
(1/24.2074)k
Â Â Â Â Â Â Â Â = 0.867502i + 0.495715j
+ 0.0413096k 
Â
The 'component calculator' calculates unknown components of 2d
vectors.
