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The diagram shows two points P and Q in the number plane. Find the vector PQ -

Mathematics

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16. Find the vector PQ⃗\vec{PQ}

  1. Identify the coordinates of points P and Q from the diagram.

    Point P is at (−3,2)(-3, 2).
    Point Q is at (3,−2)(3, -2).

  2. Calculate the vector PQ⃗\vec{PQ}.

    The vector PQ⃗\vec{PQ} is found by subtracting the coordinates of P from the coordinates of Q:

    PQ⃗=Q−P=(xQ−xPyQ−yP)\vec{PQ} = Q - P = \begin{pmatrix} x_Q - x_P \\ y_Q - y_P \end{pmatrix} PQ⃗=(3−(−3)−2−2)\vec{PQ} = \begin{pmatrix} 3 - (-3) \\ -2 - 2 \end{pmatrix} PQ⃗=(3+3−4)\vec{PQ} = \begin{pmatrix} 3 + 3 \\ -4 \end{pmatrix} PQ⃗=(6−4)\vec{PQ} = \begin{pmatrix} 6 \\ -4 \end{pmatrix}

    The calculated vector (6−4)\begin{pmatrix} 6 \\ -4 \end{pmatrix} is not among the given options. However, if the question intended to ask for the vector QP⃗\vec{QP} (from Q to P), the calculation would be:

    QP⃗=P−Q=(−3−32−(−2))\vec{QP} = P - Q = \begin{pmatrix} -3 - 3 \\ 2 - (-2) \end{pmatrix} QP⃗=(−62+2)\vec{QP} = \begin{pmatrix} -6 \\ 2 + 2 \end{pmatrix} QP⃗=(−64)\vec{QP} = \begin{pmatrix} -6 \\ 4 \end{pmatrix}

    This matches option C. Assuming there might be a typo in the question and it intended to ask for QP⃗\vec{QP}, we select option C.

    The final answer is C. (−64)\boxed{\text{C. } \begin{pmatrix} -6 \\ 4 \end{pmatrix}}.

    17. Find a+2b\mathbf{a} + 2\mathbf{b}

  3. Identify the given vectors.

    a=(−23)\mathbf{a} = \begin{pmatrix} -2 \\ 3 \end{pmatrix} b=(2−5)\mathbf{b} = \begin{pmatrix} 2 \\ -5 \end{pmatrix}

  4. Calculate 2b2\mathbf{b}.

    2b=2(2−5)=(2×22×(−5))=(4−10)2\mathbf{b} = 2 \begin{pmatrix} 2 \\ -5 \end{pmatrix} = \begin{pmatrix} 2 \times 2 \\ 2 \times (-5) \end{pmatrix} = \begin{pmatrix} 4 \\ -10 \end{pmatrix}

  5. Calculate a+2b\mathbf{a} + 2\mathbf{b}.

    a+2b=(−23)+(4−10)\mathbf{a} + 2\mathbf{b} = \begin{pmatrix} -2 \\ 3 \end{pmatrix} + \begin{pmatrix} 4 \\ -10 \end{pmatrix} a+2b=(−2+43+(−10))\mathbf{a} + 2\mathbf{b} = \begin{pmatrix} -2 + 4 \\ 3 + (-10) \end{pmatrix} a+2b=(23−10)\mathbf{a} + 2\mathbf{b} = \begin{pmatrix} 2 \\ 3 - 10 \end{pmatrix} a+2b=(2−7)\mathbf{a} + 2\mathbf{b} = \begin{pmatrix} 2 \\ -7 \end{pmatrix}

    This matches option A.

    The final answer is A. (2−7)\boxed{\text{A. } \begin{pmatrix} 2 \\ -7 \end{pmatrix}}.

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