14.5: Problems
- Page ID
- 106894
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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:
- \(\mathbf{v_1}-3\mathbf{v_2}\)
- \(\mathbf{v_3}+\frac{1}{2}\mathbf{v_4}\)
- \(\mathbf{v_1}\cdot\mathbf{v_2}\)
- \(\mathbf{v_3}\cdot\mathbf{v_4}\)
- \(\mathbf{v_1}\cdot\mathbf{v_3}\)
- \(\mathbf{v_1}\times\mathbf{v_2}\)
- \(|\mathbf{v_1}|\)
- \(|\mathbf{v_2}|\)
- \(|\mathbf{v_3}|\)
- \(|\mathbf{v_4}|\)
- \(\mathbf{\hat{v}_2}\)
- \(\mathbf{\hat{v}_4}\)
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 \(\mathbf{v_2}\).