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the resultant of two vectors a and b is perpendicular to the vector a and it's magnitude is equal to half of the magnitude of vector b .find the angle between between vector a and vector b
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the resultant of two vectors a and b is perpendicular to the vector a ...
Given: Two vectors a and b such that the resultant of a and b is perpendicular to a and its magnitude is equal to half of the magnitude of b.

To find: The angle between vector a and vector b.

Steps to solve the problem:

  • Let's assume that the angle between vector a and b is θ.

  • Given that the resultant of a and b is perpendicular to a, which means that the dot product of a and a+b is zero.

  • Therefore, a.(a+b) = 0

  • Expanding the dot product, we get a.a + a.b = 0

  • Since a.a = |a|^2, we can write the above equation as |a|^2 + a.b = 0

  • Also, given that the magnitude of the resultant is half of the magnitude of b, which means that |a+b| = |b|/2

  • Squaring both sides, we get (a+b).(a+b) = (b.b)/4

  • Expanding the dot product, we get |a|^2 + 2a.b + |b|^2 = |b|^2/4

  • Substituting |a|^2 + a.b = 0, we get a.b = -|a|^2/2

  • Substituting this value in the above equation, we get |a|^2 - |b|^2/4 = 0

  • Solving for |a|, we get |a| = |b|/2

  • Now, we can use the cosine formula to find the angle between a and b.

  • cos θ = (a.b)/(|a||b|)

  • Substituting a.b = -|a|^2/2 and |a| = |b|/2, we get cos θ = -1/2

  • Therefore, θ = 120° or 2π/3 radians.



Answer: The angle between vector a and vector b is 120° or 2π/3 radians.
Community Answer
the resultant of two vectors a and b is perpendicular to the vector a ...
120°
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