a·b=|a||b|cosθ gives projection and is used to compute work and test perpendicularity.
The dot product a·b=axbx+ayby+azbz=‖a‖‖b‖cosθ gives a scalar. It measures how much two vectors align: zero means perpendicular. Work is W=F·d and projection of a on b is (a·b)/‖b‖.
A 10 N force at 60° to a 5 m displacement does 10×5×cos60° = 25 J.
The dot product measures how much two vectors point the same way, which gives work, power and projected loads directly.
a·b=|a||b|cosθ gives projection and is used to compute work and test perpendicularity.
The dot product measures how much two vectors point the same way, which gives work, power and projected loads directly.
Their dot product is zero, which makes the cosine term vanish when both are non-zero.
Only the component of force along the displacement does work, and the dot product selects exactly that component.
Typical applications include computing the work done by a force over a displacement, finding the component of wind load normal to a panel and checking whether two beam directions are perpendicular.
Study "Vectors" first (topic 18 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
Commutative: a·b = b·a. a·a = ‖a‖². Used to find angles between members.
A 10 N force at 60° to a 5 m displacement does 10×5×cos60° = 25 J.
The next topic is "Cross product", which builds directly on this one.
It is classed as Beginner in Engineering Mathematics (topic 19 of 104). It is an entry-level topic. It relies only on basic school mathematics and the ideas introduced earlier in this discipline.
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