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Compute some sums and differences of given vectors

by RoRi

Consider the vectors in V_3

    \[ A = (1,3,6), \qquad B = (4,-3,3), \qquad C = (2,1,5). \]

Compute each of the following:

  1. A + B;
  2. A - B;
  3. A + B - C;
  4. 7A - 2B - 3C;
  5. 2A + B - C.
  1. We compute,

        \[ A+B = (1,3,6) + (4,-3,3) = (1+4,3-3, 6+3) = (5,0,9). \]

  2. We compute,

        \[ A - B = (1,3,6) - (4,-3,3) = (1-4, 3 + 3, 6 - 3) = (-3,6,3). \]

  3. We compute,

        \begin{align*} A+B - C &= (1,3,6) + (4,-3,3) - (2,1,5) \\ &= (1+4-2, 3-3-1, 6+3-5) \\ &= (3, -1, 4). \end{align*}

  4. We compute,

        \begin{align*} 7A - 2B - 3C &= 7 \cdot (1,3,6) - 2 \cdot (4,-3,3) - 3 \cdot (2,1,5) \\ &= (7,21,42) - (8, -6, 6) - (6,3,15) \\ &= (7 - 8 - 6, 21 + 6 - 3, 42 - 6 - 15) \\ &= (-7, 24, 21). \end{align*}

  5. We compute,

        \begin{align*} 2A + B - 3C &= 2 \cdot (1,3,6) + (4,-3,3) - 3 \cdot (2,1,5) \\ &= (2,6,12) + (4,-3, 3) - (6,3,15)\\ &= (2+4-6, 6-3-3, 12+3-15) \\ &= (0,0,0). \end{align*}

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