Change in the unit vectors

In summary, "Change in the unit vectors" refers to the adjustments made to the standard unit vectors in a given coordinate system, which can occur due to transformations such as rotations, reflections, or scaling. These changes can affect the representation of vectors in that space, influencing calculations and interpretations in various applications, including physics and engineering. The essence of these changes lies in understanding how the direction and magnitude of vectors are maintained or altered during such transformations.
  • #1
Manish_529
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Homework Statement
how we can write the change in unit vectors over a finite time interval, the change in unit vectors in an infinitesimally small-time interval 'dt' is given by the magnitude of the really small angle that the unit vector has turned about, but what about a change in the unit vector over a finite time interval how can we write that.
Relevant Equations
r hat= d(theta)/dt * theta hat , theta hat= - d(theta)/dt * r hat
i tried integrating the stuff but it didn't work what to do
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Last edited:
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  • #2
Please provide your actual attempt.
 
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  • #3
i have attached it
 
  • #4
i have attached the attempt please check it
 
  • #5
While your handwriting is readable, please note for future posts that attempts should be written out explicitly rather than attached as an image. Among other things, this allows quoting relevant passages etc. If you need to wrote a lot of math, refer to the forum LaTeX guide.

Your equation for ##d\hat\theta## is incorrect. It needs to be orthogonal to ##\hat\theta## or the length will change.

You can also differentiate one of the relations to decouple the equations.
 
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  • #6
can you please elaborate or just provide an attachment for your work
 
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  • #7
Manish_529 said:
can you please elaborate or just provide an attachment for your work
It is your job as the student to provide your work and ask questions regarding your understanding. Providing students with solutions to homework problems before they have solved them themselves is against PF rules.
 
  • #8
i have tried a lot but didn't really get anywhere so at least tell me what to do and where to start from
 
  • #9
I did tell you where you went wrong and you simply ignored it. If it is still unclear, please specify what is still unclear to you.
 
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  • #10
i got the thing about d theta hat but i couldn't get what to differentiate exactly
 
  • #11
Manish_529 said:
i got the thing about d theta hat but i couldn't get what to differentiate exactly
Either of the ODEs. The one for ##\hat r## or the one for ##\hat\theta##.
 
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FAQ: Change in the unit vectors

What are unit vectors?

Unit vectors are vectors that have a magnitude of one. They are used to indicate direction without specifying magnitude. In a Cartesian coordinate system, the standard unit vectors are typically represented as i, j, and k, corresponding to the x, y, and z axes, respectively.

How can a unit vector be changed?

A unit vector can be changed by multiplying it by a scalar value. If you multiply a unit vector by a scalar greater than one, it will increase in magnitude while maintaining its direction. Conversely, multiplying it by a scalar less than one will decrease its magnitude, but it will still point in the same direction.

What is the significance of changing unit vectors in physics?

Changing unit vectors is significant in physics as it allows for the representation of different quantities while maintaining direction. For instance, when calculating forces, velocities, or acceleration, unit vectors help in breaking down complex vectors into their components, making it easier to analyze and solve problems.

How do you find a unit vector from a given vector?

To find a unit vector from a given vector, you first calculate the magnitude of the vector. Then, divide each component of the vector by its magnitude. The resulting vector will have a magnitude of one, making it a unit vector. The formula is: u = v / ||v||, where u is the unit vector, v is the original vector, and ||v|| is the magnitude of vector v.

Can unit vectors be negative?

Yes, unit vectors can be negative. A negative unit vector simply indicates that the direction of the vector is opposite to the positive direction of the axis. For example, if i represents the positive x-direction, then -i represents the negative x-direction, but both have a magnitude of one.

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