What does the dot product of two vectors represent? I know how to calculate the dot product of two vectors alright. However, it is not clear to me what, exactly, does the dot product represent. The product of two numbers, 2 and 3, we say that it is 2 added to itself 3 times or something like that. But when it comes to vectors a→⋅b→, Im

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What does the dot product of two vectors represent?  I know how to calculate the dot product of two vectors alright. However, it is not clear to me what, exactly, does the dot product represent.   The product of two numbers, 2 and 3, we say that it is 2 added to itself 3 times or something like that.   But when it comes to vectors a→⋅b→, Im

Bertha Jordan · Accepted Answer

The dot product tells you what amount of one vector goes in the direction of another. For instance, if you pulled a box 10 meters at an inclined angle, there is a horizontal component and a vertical component to your force vector. So the dot product in this case would give you the amount of force going in the direction of the displacement, or in the direction that the box moved. This is important because work is defined to be force multiplied by displacement, but the force here is defined to be the force in the direction of the displacement.

lenkiklisg7 · Accepted Answer

It might help to think of multiplication of real numbers in a more geometric fashion. 2 times 3 is the length of the interval you get starting with an interval of length 3 and then stretching the line by a factor of 2.For dot product, in addition to this stretching idea, you need another geometric idea, namely projection. Imagine the line L parallel to b→ through the origin O. Now imagine projecting from the tip of the vector a→, along a line perpendicular to L, until hitting L at a point P. The dot product a→⋅b→ is the length of the line segment you get by starting with the line segment OP and then stretching the plane by a factor equal to the length of b→.I&#039;m being a little careless about plus and minus signs, but those can be incorporated into this picture too.

nick1337 · Accepted Answer

I think of dot product as the &quot;same-ness&quot; of two vectors. If two vectors are orthogonal (90∘) on one another) they are &#039;not at all the same&#039; (dot product =0), and if they are parallel they are &#039;very much the same&#039;. If you divide their dot product by the product of their magnitude, that is the argument for an arccosine function to find the angle between them. My application for the dot product is finding the angle between two vectors for calculating the force required to pull a cable through two or more pipes with a bend. It&#039;s hard to do this in a three dimensional world without knowing how to calculate the dot product.

What does the dot product of two vectors represent? I know how to

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