91Ó°ÊÓ

In Exercises 49-52, find \(\mathbf{u \cdot v}\), where \(\theta\) is the angle between \(\mathbf{u}\) and \(\mathbf{v}\). \(||\mathbf{u}|| = 4\), \(||\mathbf{v}|| = 12\), \(\theta = \dfrac{\pi}{3}\)

In Exercises 49-52, find the vector \(\mathbf{v}\) with the given magnitude and the same direction as \(\mathbf{u}\). Magnitude - ||\(\mathbf{v}\)|| \(= 8\) Direction - \(\mathbf{u} = \langle 3, 3 \rangle\)

In Exercises 53-58, determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal, parallel, or neither. \(\mathbf{u} = \langle -12, 30 \rangle\) \(\mathbf{v} = \langle \frac{1}{2}, -\frac{5}{4} \rangle\)

In Exercises 47-58, perform the operation and leave the result in trigonometric form. \(\dfrac{3(\cos\ 50^{\circ} + i\ \sin\ 50^{\circ})}{9(\cos\ 20^{\circ} + i\ \sin\ 20^{\circ})}\)

In Exercises 53-56, the initial and terminal points of a vector are given. Write a linear combination of the standard unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). Initial Point - \((-2, 1)\) Terminal Point - \((3, -2)\)

In Exercises 53-56, the initial and terminal points of a vector are given. Write a linear combination of the standard unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). Initial Point - \((0, -2)\) Terminal Point - \((3, 6)\)

In Exercises 47-58, perform the operation and leave the result in trigonometric form. \(\dfrac{(\cos\ 120^{\circ} + i\ \sin\ 120^{\circ})}{2(\cos\ 40^{\circ} + i\ \sin\ 40^{\circ})}\)

In Exercises 53-58, determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal, parallel, or neither. \(\mathbf{u} = \langle 3, 15 \rangle\) \(\mathbf{v} = \langle -1, 5 \rangle\)

DISTANCE A family is traveling due west on a road that passes a famous landmark. At a given time the bearing to the landmark is N \(62^{\circ}\)W, and after the family travels 5 miles farther the bearing is N \(38^{\circ}\)W. What is the closest the family will come to the landmark while on the road?

In Exercises 53-58, determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal, parallel, or neither. \(\mathbf{u} = \frac{1}{4}(3\mathbf{i} - \mathbf{j})\) \(\mathbf{v} = 5\mathbf{i} + 6\mathbf{j}\)