91Ó°ÊÓ

In Exercises 31-40, find the angle \(\theta\) between the vectors. \(\mathbf{u} = 2\mathbf{i} - 3\mathbf{j}\) \(\mathbf{v} = 4\mathbf{i} + 3\mathbf{j}\)

In Exercises 33-40, use Heron's Area Formula to find the area of the triangle. \(a = 3.05\), \(b = 0.75\), \(c = 2.45\)

In Exercises 31-38, find (a) \(\small{\mathbf{u}} + \small{\mathbf{v}}\), (b) \(\small{\mathbf{u}} - \small{\mathbf{v}}\), and (c) \(\small{2\mathbf{u}} - \small{3\mathbf{v}}\), Then sketch each resultant vector. \(\mathbf{u} = 2\mathbf{j}\), \(\mathbf{v} = 3\mathbf{i}\)

In Exercises 33-42, find the standard form of the complex number. Then represent the complex number graphically. \(6 \left(\cos\ \dfrac{5\pi}{12} + i\ \sin\ \dfrac{5\pi}{12} \right)\)

In Exercises 39-48, find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1. \(\mathbf{u} = \langle 3, 0 \rangle\)

In Exercises 39-44, find the area of the triangle having the indicated angle and sides. \(C\ =\ 120^{\circ}\), \(a\ =\ 4\), \(b\ =\ 6\)

In Exercises \(31-40,\) find the angle \(\theta\) between the vectors. $$ \begin{array}{l}{\mathbf{u}=\cos \left(\frac{\pi}{3}\right) \mathbf{i}+\sin \left(\frac{\pi}{3}\right) \mathbf{j}} \\ {\mathbf{v}=\cos \left(\frac{3 \pi}{4}\right) \mathbf{i}+\sin \left(\frac{3 \pi}{4}\right) \mathbf{j}}\end{array} $$

In Exercises 33-42, find the standard form of the complex number. Then represent the complex number graphically. \(7(\cos\ 0 + i\ \sin\ 0)\)

In Exercises 33-40, use Heron's Area Formula to find the area of the triangle. \(a = 1\), \(b = \frac{1}{2}\), \(c = \frac{3}{4}\)

In Exercises 39-48, find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1. \(\mathbf{u} = \langle 0, -2 \rangle\)