Key Properties of Dot Product
The dot product operation has several important mathematical properties that make it useful for vector calculations. Understanding these properties will help you master dot product calculations and apply them effectively.
Commutative Property
A·B = B·A
This property means the order of vectors doesn't matter in the dot product operation. The result is the same regardless of which vector comes first. You can see this in action in our commutative property examples.
Distributive Property
A·(B + C) = A·B + A·C
The dot product distributes over vector addition. This property is particularly useful when working with vector projections in physics applications.
Scalar Multiplication
(kA)·B = k(A·B) = A·(kB)
Scalar multiplication is associative with the dot product. This property is fundamental to understanding vector scaling in various real-world applications.
Relation to Vector Magnitude
A·A = ||A||2
The dot product of a vector with itself equals the square of its magnitude. This property is often used to calculate vector lengths in geometry problems.
Now that you understand the properties, test your knowledge with our interactive calculator or explore frequently asked questions about dot product properties.
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