# 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 $$\mathbf{v_2}$$.