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:

(2^v + 8i^v + 10j^{^}).(2^v + 2i^{^} + 3j^{^}) = ^–2*^–2 + ^–8*2 + 10*3 =4 + ^–16 + 30 = 18. Here (2^v + 8i^v + 10j^{^}) is the operator, . is the operation and (2^v + 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
i^v.j^{^} = 0
j^{^}.(^v + i^v + j^{^}) = 1
2^v. 2^{^} is ^–2*2 = ^–4
2i^v. 4i^v is ^–2*^–4 = 8
(0.577^v +0.577 i^v + 0.577j^{^}).j^{^} = 0.577
(4^{^} + 5i^{^} + 6j^{^}).(1^{^} + 2i^{^} + 3j^{^}) = 32
(4^v + 5i^v + 6j^{^}).(1^v + 2i^{^} + 3j^v) = ^–24
(6.3v + 4.7i^{^} + 5.3j^{^}).(3.9^{^} + 7.5i^v + 2.4j^v)
- = ^–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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R3 – Multiplication – Dot Product

Euclidean Geometry

Examples

Conclusion

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