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I just learned about the definition of the vector dot product and cross product.

Almost every source says that the dot product is the:

Product of the $\cos(\theta)$ component of a vector along another vector times the magnitude of the other vector.

While cross product is the:

Area of the parallelogram formed by the two vectors to be crossed.

There seems to be no meaning of what is done and why it is done so. Why do we need to find the area of the parallelogram formed when two vectors are cross produced?

It may seem as a weird question for people who have had a long bond with vectors but this question would have surely popped in the minds of everyone who is beginning the journey of vectors!

What I would like to know is the physical and a more intuitive meaning of the dot and cross products.

Kyle Oman
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Adarsh TS
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4 Answers4

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One important distinction is that dot products are scalars while cross products are vectors (or more precisely pseudovectors).

A simple physics example of a dot product would be work. $W=\vec F \cdot \Delta \vec d$. In this case the product of two vectors is a scalar. To do work there must at least some component of force in the direction of the displacement. If the angle theta between the two vectors is 90° there's no work done.

A common example of a cross product would be torque. $\vec \tau = \vec r \times \vec F$. In this case the product has a direction. Torque is different from work: if the two vectors (force and radius) are in the same direction the product is zero. Try closing a door by pushing directly towards the hinge. This produces no torque. Mathematically you can think of it as a parallelogram with zero area.

Ruslan
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M. Enns
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  • In summary, one could say that the dot product is the parallel components multiplied, while the cross product's magnitude is the perpendicular components multiplied. This is basically why these two tools are relevant in physics - so many physical phenomena, such as the work and torque this answer describes, depend purely on either parallel or perpendicular components. – Steeven May 17 '18 at 09:52
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There are many ways of answering this question - here is one which I find interesting.

If you think of vectors as columns of 3 numbers, ${\bf a}=(a_1,a_2,a_3)$ and ${\bf b} =(b_1,b_2,b_3)$, and you think about how you might multiply them together in some sensible way, you could write $\tilde{\bf a} {\bf b}$ or ${\bf a} \tilde {\bf b}$ (where the tilde means taking the transpose).

The first way gives the scalar product, and it's easy to see (by writing out the components) that $\tilde {\bf a} {\bf b}=\tilde{\bf b}{\bf a}$.

The second way - it's called the Cartesian product - gives a 3 by 3 matrix: $M_{ij}=a_ib_j$. It is not symmetric:${\bf a} \tilde {\bf b}\neq {\bf b} \tilde {\bf a}$. You can subtract the two and get $$\left(\matrix{ 0 &a_1b_2-b_1a_2&a_1b_3-b_1a_3\\a_2b_1-b_2a_1&0&a_2b_3-b_2a_3\\a_3b_1-b_3a_1&a_3b_2-b_3a_2&0}\right)$$ This is antisymmetric, so there are only 3 independent components. You can assign these to the three elements of a vector - the (1,2) matrix element goes in the (3) vector component, the (2,3) goes in (1) and (1,3) goes in (2). (Technically you do this using the Levi-Civita symbol).

The interesting point is that this only works in 3 dimensions. In 2D there is only one non-zero element, and in 4D there are 6. Scalar products are very very general, but vector products are peculiar to our 3D space.

So in terms of components the scalar product is the sum of the components multiplied together. The vector product is the difference of the two possible Cartesian products. This happens to give the area of the parallelogram, but that's just a useful spin-off.

RogerJBarlow
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Geometrically speaking, given two vectors A and B, the dot product A.B represents the measure of paralellism between the two. In other words, it is the projection that one vector has over another.

The cross product A×B on the other hand gives you the area vector of the parallelogram with lengths |A| and |B| oriented in the positive direction.

Song of Physics
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Well, in my opinion you get to see the real beauty of dot and cross product when one calculates the gradient and curl values of vector fields and functions. The physical applications are the line integrals, surface and volume integrals and the stokes's law. More importantly, the dot and cross products are extensively used in 3-D geometry.

user195911
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    One thing to note here is that curl and div aren’t ‘proper’ cross/dot products. – Jordan Abbott May 17 '18 at 10:03
  • What do you mean by a 'proper' dot/cross product? – user195911 May 17 '18 at 10:23
  • It’s more of an abuse of notation that works out for coordinates such as Cartesian. The analogy breaks down for curvilinear coordinates where taking it as a proper cross product results in losing some factors. – Jordan Abbott May 17 '18 at 11:47