Question on dot product of vectors.

In summary: , in summary, the first method is doing the dot product of the individual terms and the second method is doing the dot product of the entire equation.
  • #1
yungman
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[tex] \vec r = \hat x x + \hat y y + \hat z z \;\Rightarrow \;r = \sqrt { x^2+y^2+z^2} \;,\, \hat r= \frac { \hat x x + \hat y y + \hat z z}{ r} [/tex]

I want to find the dot product [tex] (\hat x -\hat r \frac x r) \cdot (\hat x -\hat r \frac x r) [/tex]

1) [tex] \hat x -\hat r \frac x r = \hat x - \frac { (\hat x x + \hat y y + \hat z z) x }{ r^2} = \frac { \hat x (y^2+z^2) - \hat y xy - \hat z xz}{r^2} \;\Rightarrow\; (\hat x -\hat r \frac x r) \cdot (\hat x -\hat r \frac x r) = \frac {(y^2+z^2)^2+x^2y^2+x^2z^2}{r^4}= \frac {(y^2+z^2)(x^2+y^2+z^2)}{r^4}=\frac {(y^2+z^2)}{r^2}[/tex]




2) But if I just blind do the dot product:

[tex] (\hat x -\hat r \frac x r) \cdot (\hat x -\hat r \frac x r) = 1 + \frac {x^2}{r^2} [/tex]

Here, all I did is [itex] \hat x \cdot \hat x \;\hbox { and } \hat r \cdot \hat r [/itex].




Is it true for dot product, only independent variable can dot together, [itex] \hat r[/itex] is a dependent variable of [itex] \hat x[/itex] so I cannot use the 2) method to perform dot product. Is this true?

Thanks

Alan
 
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  • #2
when u open the bracket directly (as done in point 2), you get three terms, one of x_cap.x_cap, one of r_cap.r_cap, which you have written write, but term of x_cap.r_cap you have missed.

since r_cap is (x x_cap + y y_cap + z z_cap)/[itex]\sqrt{}(x*x+y*y+z*z)[/itex], so r_cap.x_cap is not zero, as your second method is actually doing
 
  • #3
piyushkumar said:
when u open the bracket directly (as done in point 2), you get three terms, one of x_cap.x_cap, one of r_cap.r_cap, which you have written write, but term of x_cap.r_cap you have missed.

since r_cap is (x x_cap + y y_cap + z z_cap)/[itex]\sqrt{}(x*x+y*y+z*z)[/itex], so r_cap.x_cap is not zero, as your second method is actually doing

I got it. Thanks.

Alan
 

FAQ: Question on dot product of vectors.

What is the dot product of two vectors?

The dot product of two vectors is a mathematical operation that results in a scalar quantity. It is calculated by multiplying the corresponding components of the two vectors and then summing them together.

How is the dot product of vectors used in science?

The dot product of vectors is used in various fields of science, including physics, engineering, and computer science. It is used to calculate work, energy, and power in physics, as well as determining the angle between two vectors. In engineering, it is used to calculate the force exerted by a vector. In computer science, it is used in algorithms for data analysis and pattern recognition.

What is the geometric interpretation of the dot product?

The dot product of two vectors has a geometric interpretation as well. It represents the product of the magnitudes of the two vectors and the cosine of the angle between them. This means that the dot product is larger when the vectors are parallel and smaller when they are perpendicular.

What is the difference between the dot product and the cross product of vectors?

The dot product and the cross product are two different operations on vectors. The dot product results in a scalar quantity, while the cross product results in a vector. The dot product measures the similarity or alignment of two vectors, while the cross product measures the perpendicularity or rotational effect of two vectors.

How is the dot product related to vector projections?

The dot product is closely related to vector projections. The dot product of two vectors is equal to the product of the magnitude of one vector and the component of the other vector in the direction of the first vector. This is essentially the projection of one vector onto the other.

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