November 2, 2024
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R3 – Multiplication – Dot Product

Dot product multiplication in R3 uses the same approach as in classical mathematics.

The dot product of the points a (a_x, a_y, a_z) and b (b_x, b_y, b_z) is written as a \cdot b and is given by the formula where a flip (^-) precedes the Counters when the Opposite Value is ^v, i^v or j^v:

    \[a \cdot b = (^?|a_x| * ^?|b_x|) + (^?|a_y| * ^?|b_y|) + (^?|a_z| * ^?|b_z|)\]

For example:

  • (2v + 8iv + 10j^).(2v  + 2i^ + 3j^) = 2*2 + 8*2 + 10*3 =4 + 16 + 30 = 18. Here (2v + 8iv + 10j^) is the operator, . is the operation and (2v  + 2i^ + 3j^) is the operand.

Euclidean Geometry

Wikipedia describes dot product as follows: ‘Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them.’

This means that:

    \[x \cdot y = |x||y|cos\phi\]

Where |x| is the magnitude of the point, and  \phi  is the angle between two vectors x and y.  

Examples

  • j^ .^ = 0
  • j^.i^ = 0
  • j^.j^ = 1
  • iv.j^ = 0
  • j^.(v + iv + j^) = 1
  • 2v. 2^ is 2*2 = 4
  • 2iv. 4iv  is 2*4 = 8
  • (0.577v +0.577 iv  + 0.577j^).j^ = 0.577
  • (4^ + 5i^ + 6j^).(1^ + 2i^ + 3j^) = 32
  • (4v + 5iv + 6j^).(1 v  + 2i^ + 3jv) = 24
  • (6.3v + 4.7i^ + 5.3j^).(3.9^ + 7.5iv + 2.4jv)
    • = 6.3*3.9 + 4.7*7.5 + 5.3*2.4
    • = 24.57 + 35.25 + 12.72 = 72.54

Conclusion

Try these examples of dot product multiplication with our online calculator.

Finally, the output of the dot product is a Counter and can only be used as such.

Next: Cross Product

Previous: Links to Rotation and Rotication

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