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Let P and Q be points with position vectors OP = (a+2)i + j and OQ = 2ai + 5j. a) Find the vector PQ in terms of a. b) Given that PQ = 3i + 4j, find the value of a. c) Find the magnitude of PQ.

Mathematics

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Answers

3 parts
a

(a−2)i+4j(a - 2)\mathbf{i} + 4\mathbf{j}

b
5
c
5

Step-by-step solution

9 steps

Part a· 3 steps

To find PQ⃗\vec{PQ} in terms of aa, we use the formula PQ⃗=OQ⃗−OP⃗\vec{PQ} = \vec{OQ} - \vec{OP}.

  1. Substitute the given vectors.

    PQ⃗=(2ai+5j)−((a+2)i+j)\vec{PQ} = (2a\mathbf{i} + 5\mathbf{j}) - ((a+2)\mathbf{i} + \mathbf{j})

  2. Group the i\mathbf{i} and j\mathbf{j} components.

    PQ⃗=(2a−(a+2))i+(5−1)j\vec{PQ} = (2a - (a+2))\mathbf{i} + (5 - 1)\mathbf{j}

  3. Simplify the expression.

    PQ⃗=(2a−a−2)i+4j\vec{PQ} = (2a - a - 2)\mathbf{i} + 4\mathbf{j} PQ⃗=(a−2)i+4j\vec{PQ} = (a - 2)\mathbf{i} + 4\mathbf{j}

    The vector PQ⃗\vec{PQ} in terms of aa is (a−2)i+4j\boxed{(a - 2)\mathbf{i} + 4\mathbf{j}}.

Part b5

Given that PQ⃗=3i+4j\vec{PQ} = 3\mathbf{i} + 4\mathbf{j}, we equate this to the expression found in part (a).

  1. Equate the i\mathbf{i} components.

    a−2=3a - 2 = 3

  2. Solve for aa.

    a=3+2a = 3 + 2 a=5a = 5

    The value of aa is 5\boxed{5}.

Part c5

To find ∣PQ⃗∣|\vec{PQ}|, we use the given vector PQ⃗=3i+4j\vec{PQ} = 3\mathbf{i} + 4\mathbf{j} from part (b). The magnitude of a vector xi+yjx\mathbf{i} + y\mathbf{j} is given by x2+y2\sqrt{x^2 + y^2}.

  1. Substitute the components of PQ⃗\vec{PQ}.

    ∣PQ⃗∣=32+42|\vec{PQ}| = \sqrt{3^2 + 4^2}

  2. Calculate the squares.

    ∣PQ⃗∣=9+16|\vec{PQ}| = \sqrt{9 + 16}

  3. Add the values.

    ∣PQ⃗∣=25|\vec{PQ}| = \sqrt{25}

  4. Find the square root.

    ∣PQ⃗∣=5|\vec{PQ}| = 5

    The magnitude of PQ⃗\vec{PQ} is 5\boxed{5}.

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