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I couldn't get the physical significance of dot product

Qmechanic
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ankit sharma
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    Dot product measures how much two vectors are aligned with each other. Dot product is zero when the vectors are perpendicular, and has maximum magnitude when they are parallel. – Deep Jul 17 '18 at 04:20
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    Hi, welcome to Physics SE! I'm flagging this question for insufficient prior research. Can you mention some relevant things which you're familiar with in the question? –  Jul 17 '18 at 04:39
  • I have all basic knowledge and I m also able to solve all vector problem but I want to know it's physical significance – ankit sharma Aug 04 '18 at 16:39

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The dot product of two vectors $A$ and $B$ is $A\cdot B = \sum^n_{i=1} a_ib_i$ where $a_i$ and $b_i$ is the components making up the vectors. It can also be portrayed as $A\cdot B = |A||B|cos\left(\theta\right)$ where $\theta$ is the angle between the vectors and $|A|$ and $|B|$ are the magnitudes of each vector.

The physical significance is really what your application needs it to be but it is as Deep said, a relationship between how aligned two vectors are. If they are perfectly aligned, the value of the scalar product is the product of the magnitudes of the two vectors, and for less and less parallel vectors, the scalar product goes down. A lot of physical examples exists where the dot product actually is something useful, the first one that comes to mind is that with the magnetic flux being the scalar product of the magnetic field and the vector area.

If we for example have the unit vectors $u=[1,0,0]$ and $v=[0,1,0]$, making up the unit vectors in x-direction and in y-direction (if we're working with Cartesian coordinates), we get that the scalar product is $u\cdot v = 1\times0 + 0\times1 + 0\times 0 = 0$ and thus the vectors should not be aligned. If we put $u\cdot v = |u||v|cos\left(\theta\right)=0$ we can easily observe that $\theta=90^\circ$. This corresponds well to our intuitive knowledge that a vector in the $\hat{x}$ and $\hat{y}$-direction be perpendicular to one another. In the same fashion one can prove that $u\cdot u$ is aligned to eachother.

I don't know if perhaps this became to basic or not, i find it hard to set the level of my answers correctly, but i hope it helped somewhat.

DakkVader
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