Dot product multiplication in R3 uses the same approach as in classical mathematics.
The dot product of the points and
is written as
and is given by the formula where a flip (
) precedes the Counters when the Opposite Value is
or
:
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:
Where is the magnitude of the point, and
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.
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