91Ó°ÊÓ

Prove that if \(\mathbf{u}\) is a unit vector and \(\theta\) is the angle between \(\mathbf{u}\) and \(\mathbf{i},\) then \(\mathbf{u}=\cos \theta \mathbf{i}+\sin \theta \mathbf{j}.\)

Prove that if \(\mathbf{u}\) is a unit vector and \(\theta\) is the angle between \(\mathbf{u}\) and \(\mathbf{j},\) then $$\mathbf{u}=\cos \left(\frac{\pi}{2}-\theta\right) \mathbf{i}+\sin \left(\frac{\pi}{2}-\theta\right) \mathbf{j}.$$

Describe how the graph of \(g\) is related to the graph of \(f.\) $$g(x)=f(x-4)$$

Use the Law of cosines to find the angle \(\alpha\) between the vectors. (Assume \(0^{\circ} \leq \alpha \leq 180^{\circ}\) ). $$\mathbf{v}=\mathbf{i}+\mathbf{j}, \quad \mathbf{w}=2(\mathbf{i}-\mathbf{j})$$

Use the Law of cosines to find the angle \(\alpha\) between the vectors. (Assume \(0^{\circ} \leq \alpha \leq 180^{\circ}\) ). $$\mathbf{v}=3 \mathbf{i}+\mathbf{j}, \quad \mathbf{w}=2 \mathbf{i}-\mathbf{j}$$

Describe how the graph of \(g\) is related to the graph of \(f.\) $$g(x)=-f(x)$$

Represent the powers \(z, z^{2}, z^{3},\) and \(z^{4}\) graphically. Describe the pattern. $$z=\frac{\sqrt{2}}{2}(1+i)$$

Describe how the graph of \(g\) is related to the graph of \(f.\) $$g(x)=f(x)+6$$

Describe how the graph of \(g\) is related to the graph of \(f.\) $$g(x)=f(2 x)$$

Represent the powers \(z, z^{2}, z^{3},\) and \(z^{4}\) graphically. Describe the pattern. $$z=\frac{1}{2}(1+\sqrt{3} i)$$