Chapter 6

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

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 - \((-6, 4)\) Terminal Point - \((0, 1)\)

ALTITUDE The angles of elevation to an airplane from two points \(A\) and \(B\) on level ground are \(55^{\circ}\) and \(72^{\circ}\) respectively. The points \(A\) and \(B\) are 2.2 miles apart, and the airplane is east of both points in the same vertical plane. Find the altitude of the plane.

In Exercises 47-58, perform the operation and leave the result in trigonometric form. \(\dfrac{5(\cos\ 4.3 + i\ \sin\ 4.3)}{4(\cos\ 2.1 + i\ \sin\ 2.1)}\)

In Exercises 53-58, determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal, parallel, or neither. \(\mathbf{u} = \mathbf{i}\) \(\mathbf{v} = -2\mathbf{i} + 2\mathbf{j}\)

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 - \((-1, -5)\) Terminal Point - \((2, 3)\)

In Exercises 47-58, perform the operation and leave the result in trigonometric form. \(\dfrac{12(\cos\ 92^{\circ} + i\ \sin\ 92^{\circ})}{2(\cos\ 122^{\circ} + i\ \sin\ 122^{\circ})}\)

In Exercises 53-58, determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal, parallel, or neither. \(\mathbf{u} = 2\mathbf{i} - 2\mathbf{j}\) \(\mathbf{v} = -\mathbf{i} - \mathbf{j}\)

In Exercises 57-62, find the component form of \(\small{\mathbf{v}}\) and sketchthe specified vector operations geometrically, where \(\small{\mathbf{u} = 2\mathbf{i} - \mathbf{j}}\), and \(\small{\mathbf{w} = \mathbf{i} + 2\mathbf{j}}\). \(\small{\mathbf{v} = \frac{3}{2} \mathbf{u}}\)

TRUE OR FALSE? In Exercises 57-59, determine whether the statement is true or false. Justify your answer. If a triangle contains an obtuse angle, then it must be oblique.

In Exercises 47-58, perform the operation and leave the result in trigonometric form. \(\dfrac{6(\cos\ 40^{\circ} + i\ \sin\ 40^{\circ})}{7(\cos\ 100^{\circ} + i\ \sin\ 100^{\circ})}\)

In Exercises 53-58, determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal, parallel, or neither. \(\mathbf{u} = \langle \cos\ \theta\), \(\sin\ \theta \rangle\) \(\mathbf{v} = \langle \sin\ \theta\), \(-\cos\ \theta \rangle\)

In Exercises 57-62, find the component form of \(\small{\mathbf{v}}\) and sketchthe specified vector operations geometrically, where \(\small{\mathbf{u} = 2\mathbf{i} - \mathbf{j}}\), and \(\small{\mathbf{w} = \mathbf{i} + 2\mathbf{j}}\). \(\small{\mathbf{v} = \frac{3}{4} \mathbf{w}}\)

TRUE OR FALSE? In Exercises 57-59, determine whether the statement is true or false. Justify your answer. Two angles and one side of a triangle do not necessarily determine a unique triangle.

In Exercises 59-64, (a) write the trigonometric forms of the complex numbers, (b) perform the indicated operation using the trigonometric forms, and (c) perform the indicated operation using the standard forms, and check your result with that of part (b). \((2\ +\ 2i)(1\ -\ i)\)

In Exercises 59-62, find the projection of \(\mathbf{u}\) onto \(\mathbf{v}\). Then write \(\mathbf{u}\) as the sum of two orthogonal vectors, one of which is proj\(_{\mathbf{v}} \mathbf{u}\). \(\mathbf{u} = \langle 2, 2 \rangle\) \(\mathbf{v} = \langle 6, 1 \rangle\)

In Exercises 57-62, find the component form of \(\small{\mathbf{v}}\) and sketchthe specified vector operations geometrically, where \(\small{\mathbf{u} = 2\mathbf{i} - \mathbf{j}}\), and \(\small{\mathbf{w} = \mathbf{i} + 2\mathbf{j}}\). \(\small{\mathbf{v} = \mathbf{u} + 2\mathbf{w}}\)

GEOMETRY The lengths of the sides of a triangular parcel of land are approximately 200 feet, 500 feet, and 600 feet. Approximate the area of the parcel.

TRUE OR FALSE? In Exercises 57-59, determine whether the statement is true or false. Justify your answer. If three sides or three angles of an oblique triangle are known, then the triangle can be solved.

In Exercises 59-64, (a) write the trigonometric forms of the complex numbers, (b) perform the indicated operation using the trigonometric forms, and (c) perform the indicated operation using the standard forms, and check your result with that of part (b). \((\sqrt{3}\ +\ i)(1\ +\ i)\)

In Exercises 59-62, find the projection of \(\mathbf{u}\) onto \(\mathbf{v}\). Then write \(\mathbf{u}\) as the sum of two orthogonal vectors, one of which is proj\(_{\mathbf{v}} \mathbf{u}\). \(\mathbf{u} = \langle 4, 2 \rangle\) \(\mathbf{v} = \langle 1, -2 \rangle\)

In Exercises 57-62, find the component form of \(\small{\mathbf{v}}\) and sketchthe specified vector operations geometrically, where \(\small{\mathbf{u} = 2\mathbf{i} - \mathbf{j}}\), and \(\small{\mathbf{w} = \mathbf{i} + 2\mathbf{j}}\). \(\small{\mathbf{v} = -\mathbf{u} + \mathbf{w}}\)

GEOMETRY A parking lot has the shape of a parallelogram (see figure). The lengths of two adjacent sides are 70 meters and 100 meters. The angle between the two sides is \(70^{\circ}\). What is the area of the parking lot?

In Exercises 59-64, (a) write the trigonometric forms of the complex numbers, (b) perform the indicated operation using the trigonometric forms, and (c) perform the indicated operation using the standard forms, and check your result with that of part (b). \(-2i(1\ +\ i)\)

In Exercises 59-62, find the projection of \(\mathbf{u}\) onto \(\mathbf{v}\). Then write \(\mathbf{u}\) as the sum of two orthogonal vectors, one of which is proj\(_{\mathbf{v}} \mathbf{u}\). \(\mathbf{u} = \langle 0, 3 \rangle\) \(\mathbf{v} = \langle 2, 15 \rangle\)

In Exercises 57-62, find the component form of \(\small{\mathbf{v}}\) and sketchthe specified vector operations geometrically, where \(\small{\mathbf{u} = 2\mathbf{i} - \mathbf{j}}\), and \(\small{\mathbf{w} = \mathbf{i} + 2\mathbf{j}}\). \(\small{\mathbf{v} = \frac{1}{2} (3\mathbf{u} + \mathbf{w})}\)

GEOMETRY You want to buy a triangular lot measuring 510 yards by 840 yards by 1120 yards. The price of the land is $2000 per acre. How much does the land cost? (Hint: 1 acre = 4840 square yards)

In Exercises 59-64, (a) write the trigonometric forms of the complex numbers, (b) perform the indicated operation using the trigonometric forms, and (c) perform the indicated operation using the standard forms, and check your result with that of part (b). \(3i(1\ -\ \sqrt{2}i)\)

In Exercises 59-62, find the projection of \(\mathbf{u}\) onto \(\mathbf{v}\). Then write \(\mathbf{u}\) as the sum of two orthogonal vectors, one of which is proj\(_{\mathbf{v}} \mathbf{u}\). \(\mathbf{u} = \langle -3, -2 \rangle\) \(\mathbf{v} = \langle -4, -1 \rangle\)

In Exercises 57-62, find the component form of \(\small{\mathbf{v}}\) and sketchthe specified vector operations geometrically, where \(\small{\mathbf{u} = 2\mathbf{i} - \mathbf{j}}\), and \(\small{\mathbf{w} = \mathbf{i} + 2\mathbf{j}}\). \(\small{\mathbf{v} = \mathbf{u} - 2\mathbf{w}}\)

In Exercises 59-64, (a) write the trigonometric forms of the complex numbers, (b) perform the indicated operation using the trigonometric forms, and (c) perform the indicated operation using the standard forms, and check your result with that of part (b). \(\dfrac{3\ +\ 4i}{1\ -\ \sqrt{3}i}\)

In Exercises 63-66, find the magnitude and direction angle of the vector \(\mathbf{v}\). \(\small{\mathbf{v} = 6\mathbf{i} - 6\mathbf{j}}\)

TRUE OR FALSE? In Exercises 63 and 64, determine whether the statement is true or false. Justify your answer. In Heron's Area Formula, \(s\) is the average of the lengths of the three sides of the triangle.

In Exercises 59-64, (a) write the trigonometric forms of the complex numbers, (b) perform the indicated operation using the trigonometric forms, and (c) perform the indicated operation using the standard forms, and check your result with that of part (b). \(\dfrac{1\ +\ \sqrt{3}i}{6\ -\ 3i}\)

In Exercises 63-66, find the magnitude and direction angle of the vector \(\mathbf{v}\). \(\small{\mathbf{v} = -5\mathbf{i} + 4\mathbf{j}}\)

TRUE OR FALSE? In Exercises 63 and 64, determine whether the statement is true or false. Justify your answer. In addition to SSS and SAS, the Law of Cosines can be used to solve triangles with SSA conditions.

In Exercises 65 and 66, represent the powers \(z\), \(z^2\), \(z^3\), and \(z^4\) graphically. Describe the pattern. \(z\ =\ \dfrac{\sqrt{2}}{2}(1\ +\ i)\)

In Exercises 63-66, find the magnitude and direction angle of the vector \(\mathbf{v}\). \(\small{\mathbf{v} = 3(\cos 60^{\circ} \mathbf{i} + \sin 60^{\circ} \mathbf{j})}\)

WRITING A triangle has side lengths of 10 centimeters, 16 centimeters, and 5 centimeters. Can the Law of Cosines be used to solve the triangle? Explain.

In Exercises 65 and 66, represent the powers \(z\), \(z^2\), \(z^3\), and \(z^4\) graphically. Describe the pattern. \(z\ =\ \dfrac{\sqrt{1}}{2}(1\ +\ \sqrt{3}i)\)

In Exercises 63-66, find the magnitude and direction angle of the vector \(\mathbf{v}\). \(\small{\mathbf{v} = 8(\cos 135^{\circ} \mathbf{i} + \sin 135^{\circ} \mathbf{j})}\)

WRITING Given a triangle with \(b = 47\) meters, \(A = 87^{\circ}\), and \(C = 110^{\circ}\), can the Law of Cosines be used to solve the triangle? Explain.

In Exercises 67-82, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. \((1\ +\ i)^5\)

In Exercises 67-70, find two vectors in opposite directions that are orthogonal to the vector \(\mathbf{u}\). (There are many correct answers.) \(\mathbf{u} = \langle 3, 5 \rangle\)

In Exercises 67-74, find the component form of \(\mathbf{v}\) given its magnitude and the angle it makes with the positive \(x\)-axis. Sketch \(\mathbf{v}\). Magnitude - ||\(\mathbf{v}\)|| \(= 3\) Angle - \(\theta = 0^{\circ}\)

In Exercises 67-82, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. \((2\ +\ 2i)^6\)

In Exercises 67-70, find two vectors in opposite directions that are orthogonal to the vector \(\mathbf{u}\). (There are many correct answers.) \(\mathbf{u} = \langle -8, 3 \rangle\)

In Exercises 67-74, find the component form of \(\mathbf{v}\) given its magnitude and the angle it makes with the positive \(x\)-axis. Sketch \(\mathbf{v}\). Magnitude - ||\(\mathbf{v}\)|| \(= 1\) Angle - \(\theta = 45^{\circ}\)

In Exercises 67-82, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. \((-1\ +\ i)^6\)

In Exercises \(67-70,\) find two vectors in opposite directions that are orthogonal to the vector \(\mathbf{u}\) . There are many correct answers.) $$ \mathbf{u}=\frac{1}{2} \mathbf{i}-\frac{2}{3} \mathbf{j} $$