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If magnitude of sum of two unit vectors is √2 then find the magnitude of subtraction of these unit vectors?
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Most Upvoted Answer
If magnitude of sum of two unit vectors is √2 then find the magnitude ...
Unit vectors are vectors whose magnitude is equal 1.
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If magnitude of sum of two unit vectors is √2 then find the magnitude ...
Solution

Let's consider two unit vectors A and B such that


  • |A| = |B| = 1

  • |A + B| = √2



Method 1: Using dot product

We know that dot product of two unit vectors is equal to cosine of the angle between them.

Let theta be the angle between A and B.

Then,


  • A · B = cos(theta)

  • |A + B|^2 = (A + B) · (A + B)

  • |A + B|^2 = A · A + B · B + 2A · B

  • |A + B|^2 = 2 + 2cos(theta)

  • √2 = sqrt(2 + 2cos(theta))

  • cos(theta) = 1/2

  • theta = 60 degrees



Now, we can find the magnitude of the subtraction of these unit vectors using the cosine formula:


  • |A - B|^2 = |A|^2 + |B|^2 - 2|A||B|cos(theta)

  • |A - B|^2 = 2 - 2cos(theta)

  • |A - B|^2 = 1

  • |A - B| = 1



Method 2: Using vector algebra

We can write A and B in terms of their components as:


  • A = i + j + k

  • B = cos(theta)i + sin(theta)j



Using the fact that |A| = |B| = 1, we get:


  • A · A = 1

  • B · B = 1

  • A · B = cos(theta) + sin(theta)



Now, we can find the magnitude of the subtraction of these unit vectors using the vector algebra:


  • |A - B|^2 = (A - B) · (A - B)

  • |A - B|^2 = A · A + B · B - 2A · B

  • |A - B|^2 = 2 - 2cos(theta)

  • |A - B|^2 = 1

  • |A - B| = 1



Conclusion

Therefore, the magnitude of the subtraction of these unit vectors is 1.
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If magnitude of sum of two unit vectors is √2 then find the magnitude of subtraction of these unit vectors?
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