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Chemistry LibreTexts

14.5: Problems

  • Page ID
    106894
  • Problem \(\PageIndex{1}\)

    Given the following vectors in 3D:

    \[\begin{aligned} \mathbf{v_1}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}\\ \mathbf{v_2}=\frac{1}{2}\hat{\mathbf{i}}-\frac{1}{2}\hat{\mathbf{k}}\\ \mathbf{v_3}=i \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}\\ \mathbf{v_4}=-\hat{\mathbf{i}}+i \hat{\mathbf{j}}+\hat{\mathbf{k}} \end{aligned} \nonumber\]

    Calculate:

    1. \(\mathbf{v_1}-3\mathbf{v_2}\)
    2. \(\mathbf{v_3}+\frac{1}{2}\mathbf{v_4}\)
    3. \(\mathbf{v_1}\cdot\mathbf{v_2}\)
    4. \(\mathbf{v_3}\cdot\mathbf{v_4}\)
    5. \(\mathbf{v_1}\cdot\mathbf{v_3}\)
    6. \(\mathbf{v_1}\times\mathbf{v_2}\)
    7. \(|\mathbf{v_1}|\)
    8. \(|\mathbf{v_2}|\)
    9. \(|\mathbf{v_3}|\)
    10. \(|\mathbf{v_4}|\)
    11. \(\mathbf{\hat{v}_2}\)
    12. \(\mathbf{\hat{v}_4}\)

    What is the angle between \(\mathbf{v_1}\) and \(\mathbf{v_2}\)?

    Are \(\mathbf{v_3}\) and \(\mathbf{v_4}\) orthogonal?

    Write a vector orthogonal to both \(\mathbf{v_1}\) and \(\mathbf{v_2}\).

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