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Given that OP = 4j and PQ = 3i + 2j, find 2PQ, |PQ|, and the coordinates of Q.

Mathematics

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Answers

3 parts
a

6i⃗+4j⃗6\vec{i} + 4\vec{j}

b

13\sqrt{13}

c
(3, 6)

Step-by-step solution

7 steps

Here's the solution to question 1:

1) Given that OP⃗=4j⃗\vec{OP} = 4\vec{j} and PQ⃗=3i⃗+2j⃗\vec{PQ} = 3\vec{i} + 2\vec{j}

Part a· 1 step

Find 2PQ⃗2\vec{PQ}

  1. Multiply the vector PQ⃗\vec{PQ} by the scalar 2.

    2PQ⃗=2(3i⃗+2j⃗)2\vec{PQ} = 2(3\vec{i} + 2\vec{j}) 2PQ⃗=(2×3)i⃗+(2×2)j⃗2\vec{PQ} = (2 \times 3)\vec{i} + (2 \times 2)\vec{j} 2PQ⃗=6i⃗+4j⃗2\vec{PQ} = 6\vec{i} + 4\vec{j}

    The vector 2PQ⃗2\vec{PQ} is 6i⃗+4j⃗\boxed{6\vec{i} + 4\vec{j}}.

Part b

13\sqrt{13}

Find ∣PQ⃗∣|\vec{PQ}|

  1. Identify the components of vector PQ⃗\vec{PQ}.

    PQ⃗=3i⃗+2j⃗\vec{PQ} = 3\vec{i} + 2\vec{j}, so the x-component is 3 and the y-component is 2.

  2. Calculate the magnitude using the formula ∣v⃗∣=x2+y2|\vec{v}| = \sqrt{x^2 + y^2}.

    ∣PQ⃗∣=32+22|\vec{PQ}| = \sqrt{3^2 + 2^2} ∣PQ⃗∣=9+4|\vec{PQ}| = \sqrt{9 + 4} ∣PQ⃗∣=13|\vec{PQ}| = \sqrt{13}

    The magnitude of PQ⃗\vec{PQ} is 13\boxed{\sqrt{13}}.

Part c(3, 6)

Find the coordinates of Q

  1. Express the position vector of Q, OQ⃗\vec{OQ}, in terms of OP⃗\vec{OP} and PQ⃗\vec{PQ}.

    Using vector addition, OQ⃗=OP⃗+PQ⃗\vec{OQ} = \vec{OP} + \vec{PQ}.

  2. Substitute the given vectors into the equation.

    OQ⃗=4j⃗+(3i⃗+2j⃗)\vec{OQ} = 4\vec{j} + (3\vec{i} + 2\vec{j})

  3. Combine the i⃗\vec{i} and j⃗\vec{j} components.

    OQ⃗=3i⃗+(4+2)j⃗\vec{OQ} = 3\vec{i} + (4+2)\vec{j} OQ⃗=3i⃗+6j⃗\vec{OQ} = 3\vec{i} + 6\vec{j}

  4. Determine the coordinates of Q from its position vector.

    If OQ⃗=xi⃗+yj⃗\vec{OQ} = x\vec{i} + y\vec{j}, then the coordinates of Q are (x,y)(x, y).
    The coordinates of Q are (3,6)\boxed{(3, 6)}.

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