The Vector Dot Product

We learned how to add and subtract vectors, and we learned how to multiply vectors by scalars, but how can we multiply two vectors together? There are two ways to do this, and one of them is called the dot product. Let's learn how to find the dot product of two vectors now!
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[@TheRojo387]

Btw Dave, I have cracked vector rotation, relying entirely on scalar and dot products. My solve can also be extended to higher dimensions, as I also standardise with the right-hand rule. I also note that you can find the angle by looking for one co-ordinate on a unit circle, and measure the area beneath it and add the triangle adjacent to it and with its opposite point at the origin, and multiply all by two, to get the answer in radians. But if degrees are your fancy, you gotta multiply instead by 90/π! Or…you could just use a protractor. Et voilà; no calculator needed!

[@ItCanAlwaysGetWorse]

Wonderful explanation. Very clear and succinct. I would only add the 2d visual notion that "vector dot product" (of two vectors) is obtained by multiplying the length of either vector by the "projection" of one vector over the other (think of laying the angles vectors flat on a table, and having a lamp illuminating from "above". You would multiply the length of the vector laying flat by the length of the shadow of the companion.).

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[@styxoohhh]

I thought that the dot product is always positive because the result is in absolute value, as written our module. Maybe there is just a problem in the publishing of our modules. Thank you for this video, all things are clarified. 💯

[@carultch]

The dot product is always a scalar, but it is not always positive. Dot product means multiply corresponding components, and add up the various results. If some of those components are negative, you very well could get a negative dot product.

[@anikasmathmagic4105]

At around 3:31 why did we have to take everything under a square root to find the length of |a| ?

[@lafudge2929]

I would try to give an explanation of how a double pythagorean theorem is used to obtain the length of a 3-dimensional vector, but without proper formatting and diagrams, I think that'd hurt more than it'd help. Instead, try reading this: mathinsight.org/vectors_cartesian_coordinates_2d_3d

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