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Let a = i + j and b = j + k . The vector b is rotated by an acute angle in the plane containing a and b such that it becomes perpendicular to a . If the new vector obtained after rotation is c and w = 4 i - j + 5 k , then the value of 3 ( c w ) is equal to:

Options

  1. A-5
  2. B5 3
  3. C-5 3
  4. D5

Correct answer

D. 5

Step-by-step solution

Since c lies in the plane of a and b , it can be expressed as a linear combination: c = a + b = ( i + j ) + ( j + k ) = i + ( + ) j + k We are given that c is perpendicular to a : c a = 0 ( )(1) + ( + )(1) + ( )(0) = 0 2 + = 0 = -2 Substituting = -2 into the expression for c : c = ( i - j - 2 k ) A pure rotation preserves the magnitude of the vector, so | c | = | b | = 1^2 + 1^2 = 2 . | c | = | | 1^2 + (-1)^2 + (-2)^2 = | | 6 | | 6 = 2 | | = 1 3 Thus, c = 1 3 ( i - j - 2 k ) . To determine the correct sign, we use

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