Vectors
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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.  

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:
 A = 1
     2
     3
 B = -2
     -3
      6
C = 2
D = 3
are given by: (INTERNAL DIVISION)
 X = [CB + DA] / (C+D)
   = -0.2
        0
      4.2
and: (EXTERNAL DIVISION)
[CB - DA] / (C-D) = 7
                   12
                   -3
The midpoint of the line segment AB is given by:
 [A + B] / 2 = -0.5
               -0.5
                4.5

 

Vector Editor

Edit the five vector variables in this tab.

2d or 3d Vector Calculations

 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:
   A + B = (xA + xB)i + (yA + yB)j + (zA + zB)k
         = (1 + -2)i
           + (2 + -3)j
            + (3 + 6)k
         = -1i + -1j + 9k
 |A + B| = 9.11043,

Vector Difference:
   A - B = (xA - xB)i + (yA - yB)j + (zA - zB)k
         = (1 - -2)i
           + (2 - -3)j
            + (3 - 6)k
         = 3i + 5j + -3k
 |A - B| = 6.55744,

Scalar ('dot') product:
     A·B = (xA × xB) + (yA × yB) + (zA × zB)
         = (1 × -2)
           + (2 × -3)
            + (3 × 6)
         = 10

Angle between vectors A and B:
     A·B = |A||B|cos(T), where 'T' is the
     angle between 'A' and 'B',
        So T = acos[ A·B / (|A||B|) ]
             = acos[ 10 / (3.74166 × 7) ]
             = 67.5547°

Vector ('cross') product:
     A×B = [(yA × zB) - (zA × yB)]i
          + [(zA × xB) - (xA × zB)]j
            + [(xA × yB) - (yA × xB)]k
         = [(2 × 6) - (3 × -3)]i
          + [(3 × -2) - (1 × 6)]j
            + [(1 × -3) - (2 × -2)]k
         = 21i + -12j + 1k

Area of parallelogram and perpendicular unit vector:
 A×B = u|A||B|sin(T), where 'u' is a unit vector
  perpendicular to 'A' and 'B', and 'T' is the angle
  between 'A' and 'B'.
 The scalar expression: |A||B|sin(T) = |A×B|
  represents the area of the parallelogram
  defined by 'A' and 'B'.
  In this case, |A×B| = 24.2074
  So 'u' = [1/|A×B|](A×B)
         = (21/24.2074)i
           + (-12/24.2074)j
            + (1/24.2074)k
         = 0.867502i + -0.495715j + 0.0413096k
 

The 'component calculator' calculates unknown components of 2d vectors.