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Q & A: vector products

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Most recent answer: 11/15/2014
Dot and cross product of vectors in real life? not my homework, I was just reading my highschool book and about vectors and I cant help but wondering how two vector quantities multiplied can give a scalar. I know that there are examples but how does it work. Also in real life example howdo you know when vectors are multiplied and when they are added? Plane goes north 300 km then west 500 km, in this case the vectors are added and the resultant is calculated by addition. Now give me one real life example of when we have to calculate by dot product and when by cross product. Thanks
- Shahab khan (age 18)
karachi, pakistan

Vector cross products are a rather specialized tool. They appear in some extremely important physical laws, especially Maxwell's equations for electromagnetism. You can read elsewhere about those. Since the power to run the computer you're using now is probably coming from a generator based on Maxwell's equations, it's a very real-life application. I just a few minutes ago answered a question from Nepal about how magnets can heat up a coil. The current in the coil is driven by a field that's the cross product of the magnetic field and the relative velocity of the coil and the magnet.

Vector dot products are a much more general tool. The dot product tells us whether two vectors are similar to each other (positive product), opposite (negative), or at about right angles (zero). If you want to know how much a force is changing the energy of a moving object, that power depends on the dot product of the force and the object's velocity. If the object is in a circular orbit, those vectors are at right angles so the dot product is zero and the energy doesn't change.

The dot product isn't limited to these ordinary vectors in 3-D space. It's used for all sorts of more abstract vectors too. For example in statistical predictions, people use regression methods. You might, for example, predict someone's weight based on their height in an approximate linear model. The coefficient (slope) saying how much extra weight is predicted for each increase of height is obtained from a vector dot product, where the two vectors are the lists of everyone's heights and weights. This dot product is then divided by the square root of the dot product of the height vector with itself, giving the weight per height slope.  Once you learn vector dot products it's very hard to imagine thinking of the world without using them.

Mike W.

(published on 11/15/2014)

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