14.5: Problems

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 \(\mathbf{v_2}\).