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a =2 i + j - k , b = i - j and c =5 i - j + k , then unit vector parallel to a + b - c but in opposite direction is

Options

  1. A1 3 (2 i - j -2 k )
  2. B1 3 (2 i - j +2 k )
  3. CNone of these
  4. D1 2 (2 i - j +2 k )

Correct answer

B. 1 3 (2 i - j +2 k )

Step-by-step solution

Given vectors a = 2 i + j - k , b = i - j , and c = 5 i - j + k . Calculate the vector v = a + b - c : v = (2 i + j - k ) + ( i - j ) - (5 i - j + k ) v = (2 + 1 - 5) i + (1 - 1 + 1) j + (-1 - 0 - 1) k v = -2 i + j - 2 k The magnitude of v is | v | = (-2)^2 + (1)^2 + (-2)^2 = 4 + 1 + 4 = 9 = 3 . The unit vector in the direction of v is u = v | v | = 1 3 (-2 i + j - 2 k ) . The unit vector in the opposite direction is - u = - 1 3 (-2 i + j - 2 k ) = 1 3 (2 i - j + 2 k ) . Answer: 1 3 (2 i - j +2 k )

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