Skip to main content
LibreTexts - Ukrayinska

14.5: Проблеми

  • Page ID
    18207
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    Проблема\(\PageIndex{1}\)

    Задано наступні вектори в 3D:

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

    Розрахувати:

    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. \(\mathbf{\hat{v}_4}\)

    Що таке кут між\(\mathbf{v_1}\) і\(\mathbf{v_2}\)?

    Є\(\mathbf{v_3}\) і\(\mathbf{v_4}\) ортогональні?

    Напишіть вектор, ортогональний обох\(\mathbf{v_1}\) і\(\mathbf{v_2}\).