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## Physics library

### Course: Physics library>Unit 13

Lesson 3: Electric motors

# Dot vs. cross product

This passage discusses the differences between the dot product and the cross product. While both involve multiplying the magnitudes of two vectors, the dot product results in a scalar quantity, which indicates magnitude but not direction, while the cross product results in a vector, which indicates magnitude and direction. The author also explains how to use the right hand rule to determine the direction of the cross product vector. Created by Sal Khan.

## Want to join the conversation?

• Why are they multiplied though? What does |a||b|cos(theta) really represent conceptually? I see what they are and what the difference between them is, but I don't see why they should be multiplied rather than, say, added.
• Hi John, when dealing with vectors, the orientation in the coordinate system and among each other are decisive factors, that you don't have when multiplying plain scalar numbers.
With the two kinds of multiplication of vectos, the projection of one to the other is included.
Taking, for example, two parallel vectors: the dot product will result in cos(0)=1 and the multiplication of the vector lengths, whereas the cross product will produce sin(0)=0 and zooms down all majesty of the vectors to zero.

Another difference is the result of the calculation: Sal showed, that you're getting a plain SCALAR (number) as a result of the dot product, whereas the cross product only really makes sense in the 3d-space, because the resulting VECTOR is perpendicular to both other vectors (right hand rule). Try that in 2 dimensions ;-)
• can anyone please tell me the difference between precision and accuracy?
• Accuracy is about how close you get to the true value; if the actual answer is 4 and you get 3.9 then that's more accurate than 9.6; it's closer to 4. Precision is to do with the spread of your values; the more closely grouped the values you get, the more precise they are. For example, values of 8.9, 8.8, 8.9, 8.7, 8.8 are more PRECISE than 3.6, 4.7, 5.3, 2.6, 4.2 but the second set of values are more ACCURATE as they are closer to 4 on average.

I hope this made some sense to you!
• That was very useful, thank you. But if i may ask, when do i use the dot product and when do i use the cross product? they almost have opposite meanings. another thing In physics when we multiply 2 forces we just, for example do 10X8 and that's it. and what exactly is the vector projection, is it the "shadow" as you referred to it?
• why if 2 vectors perpendicular to each other are crossed do I get a vector orthogonal to both of em?? maybe because, if I were compressing two strong steel rods mutually perpendicular with a supermassive force, they would rather bend into the 3rd dimension tha the two forces' resultant :D
• A good example is gyroscopic force. Vectors that work together to cause a disk to spin create an orthoganal force that resists change.
(1 vote)
• Do you, after explaining the cross product, you just use a determinant as a shortcut in later videos? That would be very simple to use for later, once you know the actual reasoning.
• I shall be optimistic and ask: To what shortcut do you refer?
The determinant I know requires the same number of steps because it computes the same function. So let's define the cross product as a determinant, right now.
{Wave magic wand}
Am I missing a shortcut? What step can I leave out?

I'm not making fun. Such a shortcut would be very useful to me. I have hunted for one. But I think the answer is in the question; the determinant constructs each term by definition. No mathematical shortcuts. Am I missing something?
(1 vote)
• While finding the dot product, why is it that the final quantity doesn't have any direction?
• I learned in school about a different method of the dot product. That is:
Two vectors- x <a1,b1> and y <a2, b2>

and the dot product of x dot y was a1a2+b1b2

Is this also another correct way to do the dot product?
Thanks
• Oh okay, Thank you for clarifying!
(1 vote)
• Is the product of two similar vectors(say, velocity) always a dot product?
I mean, can a 'vector a' when multiplied with itself yeild a cross product?
• It won't. The result would be zero, since the sine of 90 degrees is zero.
(1 vote)
• If a is opposite in direction to b, in which direction will the n be in? I tried to do it with the right hand rule but it can go both ways.
(1 vote)
• It´s actually 0, the direction is the same, you just change the orientation.
The more rigorous way to see it is with the sine , sin(0)=0 and sin(180º) are both 0, so operating you get that n is 0, no direction.