Chapter 6

The ________ ________ of a complex number \(a + bi\) is the distance between the origin \((0, 0)\) and the point \((a, b)\).

The ________ ________ of two vectors yields a scalar, rather than a vector.

A ________ ________ ________ can be used to represent a quantity that involves both magnitude and direction.

If you are given three sides of a triangle, you would use the Law of ________ to find the three angles of the triangle.

The dot product of \(\mathbf{u} = \langle \textit{u}_1, \textit{u}_2 \rangle\) and \(\mathbf{v} = \langle \textit{v}_1, \textit{v}_2 \rangle\) is \(\mathbf{u} \cdot \mathbf{v}\ =\) ________ .

The directed line segment \(\overset{\rightharpoonup} {\small PQ}\) has ________ point \(\small P\) and ________ point \(\small Q\).

If you are given two angles and any side of a triangle, you would use the Law of ________ to solve the triangle.

For triangle \(ABC\), the Law of Sines is given by \(\dfrac{a}{sin\ A}\ =\) ___________ \(=\ \dfrac{c}{sin\ C}\).

________ Theorem states that if \(z = r(\cos\ \theta + i \sin\ \theta)\) is a complex number and \(n\) is a positive integer, then \(z^{n} = r^{n}(\cos\ n\theta + i \sin\ n\theta)\).

If \(\theta\) is the angle between two nonzero vectors \(\mathbf{u}\) and \(\mathbf{v}\), then \(\cos\ \theta =\) ________ .

The ________ of the directed line segment \(\overset{\rightharpoonup} {\small PQ}\) is denoted by \(\parallel \overset{\rightharpoonup} {\small PQ} \parallel\).

The standard form of the Law of Cosines for \(\cos\ B\ =\ \dfrac{a^2 + c^2 - b^2}{2ac}\) is ________ .

The complex number \(u = a + bi\) is an ________ ________ of the complex number \(z\) if \(z = u^{n} = (a + bi)^{n}\).

The vectors \(\mathbf{u}\) and \(\mathbf{v}\) are ________ if \(\mathbf{u \cdot v} = 0\).

The set of all directed line segments that are equivalent to a given directed line segment \(\overset{\rightharpoonup} {\small PQ}\) is a ________ \(\small{\mathbf{v}}\) in the plane.

The Law of Cosines can be used to establish a formula for finding the area of a triangle called ________ ________ Formula.

In Exercises 5-10, plot the complex number and find its absolute value. \(-6 + 8i\)

In order to show that two vectors are equivalent, you must show that they have the same ________ and the same ________ .

In Exercises 5-10, plot the complex number and find its absolute value. \(5 - 12i\)

The directed line segment whose initial point is the origin is said to be in ________ ________ .

In Exercises 5-10, plot the complex number and find its absolute value. \(-7i\)

In Exercises 7-14, find the dot product of \(\mathbf{u}\) and \(\mathbf{v}\). \(\mathbf{u} = \langle 7, 1 \rangle\) \(\mathbf{v} = \langle -3, 2 \rangle\)

A vector that has a magnitude of 1 is called a ________ ________ .

In Exercises 5-10, plot the complex number and find its absolute value. \(-7\)

In Exercises 7-14, find the dot product of \(\mathbf{u}\) and \(\mathbf{v}\). \(\mathbf{u} = \langle 6, 10 \rangle\) \(\mathbf{v} = \langle -2, 3 \rangle\)

The two basic vector operations are scalar ________ and vector ________ .

In Exercises 5-10, plot the complex number and find its absolute value. \(4 - 6i\)

In Exercises 7-14, find the dot product of \(\mathbf{u}\) and \(\mathbf{v}\). \(\mathbf{u} = \langle -4, 1 \rangle\) \(\mathbf{v} = \langle 2, -3 \rangle\)

The vector \(\small{\mathbf{u} + \mathbf{v}}\) is called the ________ of vector addition.

In Exercises 5-20, use the Law of Cosines to solve the triangle. Round your answers to two decimal places. \(a = 11\), \(b = 15\), \(c = 21\)

In Exercises 5-24, use the Law of Sines to solve the triangle.Round your answers to two decimal places. \(A\ =\ 102.4^{\circ}\), \(C\ =\ 16.7^{\circ}\), \(a\ =\ 21.6\)

In Exercises 5-10, plot the complex number and find its absolute value. \(-8 + 3i\)

In Exercises 7-14, find the dot product of \(\mathbf{u}\) and \(\mathbf{v}\). \(\mathbf{u} = \langle -2, 5 \rangle\) \(\mathbf{v} = \langle -1, -8 \rangle\)

The vector sum \(\small{v_1 \mathbf{i} + v_2 \mathbf{j}}\) is called a ________ ________ of the vectors \(\small{\mathbf{i}}\) and \(\small{\mathbf{j}}\), and the scalars \(v_1\) and \(v_2\) are called the ________ and ________ components of \(\small{\mathbf{v}}\), respectively.

In Exercises 5-20, use the Law of Cosines to solve the triangle. Round your answers to two decimal places. \(a = 55\), \(b = 25\), \(c = 72\)

In Exercises 5-24, use the Law of Sines to solve the triangle.Round your answers to two decimal places. \(A\ =\ 24.3^{\circ}\), \(C\ =\ 54.6^{\circ}\), \(c\ =\ 2.68\)

In Exercises 7-14, find the dot product of \(\mathbf{u}\) and \(\mathbf{v}\). \(\mathbf{u} = 4\mathbf{i} - 2\mathbf{j}\) \(\mathbf{v} = \mathbf{i} - \mathbf{j}\)

In Exercises 5-20, use the Law of Cosines to solve the triangle. Round your answers to two decimal places. \(a = 75.4\), \(b = 52\), \(c = 52\)

In Exercises 7-14, find the dot product of \(\mathbf{u}\) and \(\mathbf{v}\). \(\mathbf{u} = 3\mathbf{i} + 4\mathbf{j}\) \(\mathbf{v} = 7\mathbf{i} - 2\mathbf{j}\)

In Exercises 5-20, use the Law of Cosines to solve the triangle. Round your answers to two decimal places. \(a = 1.42\), \(b = 0.75\), \(c = 1.25\)

In Exercises 7-14, find the dot product of \(\mathbf{u}\) and \(\mathbf{v}\). \(\mathbf{u} = 3\mathbf{i} + 2\mathbf{j}\) \(\mathbf{v} = -2\mathbf{i} - 3\mathbf{j}\)

In Exercises 5-20, use the Law of Cosines to solve the triangle. Round your answers to two decimal places. \(A = 120^{\circ}\), \(b = 6\), \(c = 7\)

In Exercises 5-24, use the Law of Sines to solve the triangle.Round your answers to two decimal places. \(A\ =\ 35^{\circ}\), \(C\ =\ 65^{\circ}\), \(c\ =\ 10\)

In Exercises 7-14, find the dot product of \(\mathbf{u}\) and \(\mathbf{v}\). \(\mathbf{u} = \mathbf{i} - 2\mathbf{j}\) \(\mathbf{v} = -2\mathbf{i} + \mathbf{j}\)

In Exercises 15-32, represent the complex number graphically, and find the trigonometric form of the number. \(1 + i\)

In Exercises 15-24, use the vectors \(\mathbf{u} = \langle 3, 3 \rangle\), \(\mathbf{v} = \langle -4, 2 \rangle\), and \(\mathbf{w} = \langle 3, -1 \rangle\) to find the indicated quantity. State whether the result is a vector or a scalar. \(\mathbf{u} \cdot \mathbf{u}\)

In Exercises 5-20, use the Law of Cosines to solve the triangle. Round your answers to two decimal places. \(B = 10^{\circ}35'\), \(a = 40\), \(c = 30\)

In Exercises 5-24, use the Law of Sines to solve the triangle.Round your answers to two decimal places. \(A\ =\ 55^{\circ}\), \(B\ =\ 42^{\circ}\), \(c\ =\ \frac{3}{4}\)

In Exercises 15-32, represent the complex number graphically, and find the trigonometric form of the number. \(5 - 5i\)

In Exercises 15-24, use the vectors \(\mathbf{u} = \langle 3, 3 \rangle\), \(\mathbf{v} = \langle -4, 2 \rangle\), and \(\mathbf{w} = \langle 3, -1 \rangle\) to find the indicated quantity. State whether the result is a vector or a scalar. \(3\mathbf{u} \cdot \mathbf{v}\)