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Given that A (2;2), B (0; k) and C (3; 2) are points on a plane, relative to the origin, O, find, (a) OA giving the answer in surd form, (b) BC in terms of k, (c) k given that OA is parallel to BC.

Mathematics

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Answers

3 parts
a

222\sqrt{2}

b

(32−k)\begin{pmatrix} 3 \\ 2 - k \end{pmatrix}

c
-1

Step-by-step solution

9 steps

Part a· 3 steps

  1. Write the position vector for point A.

    OA⃗=(22)\vec{OA} = \begin{pmatrix} 2 \\ 2 \end{pmatrix}

  2. Calculate the magnitude of OA⃗\vec{OA} using the distance formula.

    ∣OA⃗∣=22+22=4+4=8|\vec{OA}| = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8}

  3. Simplify the surd.

    8=4×2=22\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}

    The magnitude of OA⃗\vec{OA} is 22\boxed{2\sqrt{2}}.

Part b

(32−k)\begin{pmatrix} 3 \\ 2 - k \end{pmatrix}

  1. Write the position vectors for points B and C.

    OB⃗=(0k),OC⃗=(32)\vec{OB} = \begin{pmatrix} 0 \\ k \end{pmatrix}, \quad \vec{OC} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}

  2. Calculate the vector BC⃗\vec{BC} using BC⃗=OC⃗−OB⃗\vec{BC} = \vec{OC} - \vec{OB}.

    BC⃗=(32)−(0k)=(3−02−k)=(32−k)\vec{BC} = \begin{pmatrix} 3 \\ 2 \end{pmatrix} - \begin{pmatrix} 0 \\ k \end{pmatrix} = \begin{pmatrix} 3 - 0 \\ 2 - k \end{pmatrix} = \begin{pmatrix} 3 \\ 2 - k \end{pmatrix}

    The vector BC⃗\vec{BC} is (32−k)\boxed{\begin{pmatrix} 3 \\ 2 - k \end{pmatrix}}.

Part c-1

  1. Use the condition that OA⃗\vec{OA} is parallel to BC⃗\vec{BC}.

    This means OA⃗=λBC⃗\vec{OA} = \lambda \vec{BC} for some scalar λ\lambda.

    (22)=λ(32−k)\begin{pmatrix} 2 \\ 2 \end{pmatrix} = \lambda \begin{pmatrix} 3 \\ 2 - k \end{pmatrix}

  2. Equate the corresponding components to form equations.

    2=3λ(1)2 = 3\lambda \quad \text{(1)} 2=λ(2−k)(2)2 = \lambda(2 - k) \quad \text{(2)}

  3. Solve equation (1) for λ\lambda.

    λ=23\lambda = \frac{2}{3}

  4. Substitute the value of λ\lambda into equation (2) and solve for kk.

    2=23(2−k)2 = \frac{2}{3}(2 - k) 3×2=2(2−k)3 \times 2 = 2(2 - k) 6=4−2k6 = 4 - 2k 2k=4−62k = 4 - 6 2k=−22k = -2 k=−1k = -1

    The value of kk is −1\boxed{-1}.

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