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Given vectors OA = 2p, OQ = q, Q is the midpoint of OC, OABP is a straight line, and OA = AB = BP. Express OB, BC, AQ, and CP in terms of p and/or q.

Mathematics

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6 steps

Given:

  • OA⃗=2p⃗\vec{OA} = 2\vec{p}
  • OQ⃗=q⃗\vec{OQ} = \vec{q}
  • Q is the midpoint of OC, so OC⃗=2OQ⃗=2q⃗\vec{OC} = 2\vec{OQ} = 2\vec{q}.
  • OABP is a straight line.
  • From the diagram, OA = AB = BP. Since OA⃗=2p⃗\vec{OA} = 2\vec{p}, it follows that AB⃗=2p⃗\vec{AB} = 2\vec{p} and BP⃗=2p⃗\vec{BP} = 2\vec{p}.

Part a· 1 step

Express OB⃗\vec{OB} in terms of p⃗\vec{p} and/or q⃗\vec{q}.

  1. Use the vector addition rule OB⃗=OA⃗+AB⃗\vec{OB} = \vec{OA} + \vec{AB}.

    OB⃗=2p⃗+2p⃗\vec{OB} = 2\vec{p} + 2\vec{p} OB⃗=4p⃗\vec{OB} = 4\vec{p}

    The final answer is \boxed{\text{4\vec{p}}}.

Part b· 1 step

Express BC⃗\vec{BC} in terms of p⃗\vec{p} and/or q⃗\vec{q}.

  1. Use the vector addition rule BC⃗=BO⃗+OC⃗\vec{BC} = \vec{BO} + \vec{OC}.

    We know BO⃗=−OB⃗=−4p⃗\vec{BO} = -\vec{OB} = -4\vec{p} and OC⃗=2q⃗\vec{OC} = 2\vec{q}.

    BC⃗=−4p⃗+2q⃗\vec{BC} = -4\vec{p} + 2\vec{q}

    The final answer is \boxed{\text{-4\vec{p} + 2\vec{q}}}.

Part c· 1 step

Express AQ⃗\vec{AQ} in terms of p⃗\vec{p} and/or q⃗\vec{q}.

  1. Use the vector addition rule AQ⃗=AO⃗+OQ⃗\vec{AQ} = \vec{AO} + \vec{OQ}.

    We know AO⃗=−OA⃗=−2p⃗\vec{AO} = -\vec{OA} = -2\vec{p} and OQ⃗=q⃗\vec{OQ} = \vec{q}.

    AQ⃗=−2p⃗+q⃗\vec{AQ} = -2\vec{p} + \vec{q}

    The final answer is \boxed{\text{-2\vec{p} + \vec{q}}}.

Part d· 3 steps

Express CP⃗\vec{CP} in terms of p⃗\vec{p} and/or q⃗\vec{q}.

  1. Use the vector addition rule CP⃗=CO⃗+OP⃗\vec{CP} = \vec{CO} + \vec{OP}.

    We know CO⃗=−OC⃗=−2q⃗\vec{CO} = -\vec{OC} = -2\vec{q}.

  2. Calculate OP⃗\vec{OP}.

    OP⃗=OA⃗+AB⃗+BP⃗\vec{OP} = \vec{OA} + \vec{AB} + \vec{BP} OP⃗=2p⃗+2p⃗+2p⃗\vec{OP} = 2\vec{p} + 2\vec{p} + 2\vec{p} OP⃗=6p⃗\vec{OP} = 6\vec{p}

  3. Substitute CO⃗\vec{CO} and OP⃗\vec{OP} into the expression for CP⃗\vec{CP}.

    CP⃗=−2q⃗+6p⃗\vec{CP} = -2\vec{q} + 6\vec{p}

    The final answer is \boxed{\text{6\vec{p} - 2\vec{q}}}.

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