Chapter 6

When you are given three sides of a triangle, you use the Law of _____ to find the three angles of the triangle.

The __________ ____________ of a complex number \(a+b i\) is the distance between the origin \((0,0)\) and the point \((a, b)\)

\(A\)___________- can be used to represent a quantity that involves both magnitude and direction.

Fill in the blanks. The _____ _____ of two vectors yields a scalar, rather than a vector.

Vocabulary: Fill in the blanks. An_____triangle is a triangle that has no right angle.

When you are given two angles and any side of a triangle, you use the Law of _____ to solve the triangle.

The ___________ ___________ of a complex number \(z=a+b i\) is given by z \(z=r(\cos \theta+i \sin \theta),\) where \(r\) is the ___________ of \(z\) and \(\theta\) is the ___________ of \(z\).

Fill in the blanks. The dot product of \(\mathbf{u}=\left\langle u_{1}, u_{2}\right\rangle\) and \(\mathbf{v}=\left\langle v_{1}, v_{2}\right\rangle\) is \(\mathbf{u} \cdot \mathbf{v}=\) _____.

The directed line segment \(\vec{P Q}\) has___________point \(P\) and________point \(Q\)

Vocabulary: Fill in the blanks. For triangle \(A B C,\) the Law of Sines is \(\frac{a}{\sin A}=\)______ \(=\frac{c}{\sin C}\)

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

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

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

The____________of the directed line segment \(\vec{P Q}\) is denoted by \(\|\vec{P Q}\|\)

Vocabulary: Fill in the blanks. Two ________ and one ________ determine a unique triangle.

The complex number \(u=a+b i\) is an __________ ___________ of the complex number \(z\) when \(z=u^{n}=(a+b i)^{n}\)

Fill in the blanks. The vectors \(\mathbf{u}\) and \(\mathbf{v}\) are _____ when \(\mathbf{u} \cdot \mathbf{v}=0\)

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

The set of all directed line segments that are equivalent to a given directed line segment \(\vec{P Q}\) is a__________\(\mathbf{v}\) in the plane.

Vocabulary: Fill in the blanks. The area of an oblique triangle is \(\frac{1}{2} b c \sin A=\frac{1}{2} a b \sin C=\) ________.

Finding the Absolute Value of a Complex Number In Exercises \(5-10,\) plot the complex number and find its absolute value. $$-6+8 i$$

Fill in the blanks. The projection of \(\mathbf{u}\) onto \(\mathbf{v}\) is given by \(\mathrm{proj}_{\mathrm{v}} \mathrm{u}=\)_____.

In order to show that two vectors are equivalent, you must show that they have the same__________and the same__________

Finding the Absolute Value of a Complex Number In Exercises \(5-10,\) plot the complex number and find its absolute value. $$5-12 i$$

The directed line segment whose initial point is the origin is said to be in________

In Exercises \(7-14,\) find \(\mathbf{u} \cdot \mathbf{v}\) $$\mathbf{u}=\langle 7,1\rangle$$ $$\mathbf{v}=\langle- 3,2\rangle$$

Finding the Absolute Value of a Complex Number In Exercises \(5-10,\) plot the complex number and find its absolute value. $$-7 i$$

A vector that has a magnitude of 1 is called a___________

In Exercises \(7-14,\) find \(\mathbf{u} \cdot \mathbf{v}\) $$\mathbf{u}=\langle 6,10\rangle$$ $$\mathbf{v}=\langle- 2,3\rangle$$

The two basic vector operations are scalar___________and vector_________

In Exercises \(7-14,\) find \(\mathbf{u} \cdot \mathbf{v}\) $$\mathbf{u}=\langle- 4,1\rangle$$ $$\mathbf{v}=\langle 2,-3\rangle$$

Finding the Absolute Value of a Complex Number In Exercises \(5-10,\) plot the complex number and find its absolute value. $$4-6 i$$

In Exercises \(7-14,\) find \(\mathbf{u} \cdot \mathbf{v}\) $$\mathbf{u}=\langle- 2,5\rangle$$ $$\mathbf{v}=\langle- 1,-8\rangle$$

Finding the Absolute Value of a Complex Number In Exercises \(5-10,\) plot the complex number and find its absolute value. $$-8+3 i$$

The vector sum \(v_{1} \mathbf{i}+v_{2} \mathbf{j}\) is called a___________of the vectors i and j, and the scalars \(v_{1}\) and\(v_{2}\) are called the________and_____________components of \(\mathbf{v},\) respectively.

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places. $$A=24.3^{\circ}, \quad C=54.6^{\circ}, \quad c=2.68$$

In Exercises \(7-14,\) find \(\mathbf{u} \cdot \mathbf{v}\) $$\mathbf{u}=4 \mathbf{i}-2 \mathbf{j}$$ $$\mathbf{v}=\mathbf{i}-\mathbf{j}$$

Trigonometric Form of a Complex Number \(\mathrm{In}\) Exercises \(11-30\) , represent the complex number graphically. Then write the trigonometric form of the number. $$1+i$$

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places. $$A=83^{\circ} 20^{\prime}, \quad C=54.6^{\circ}, \quad c=18.1$$

In Exercises \(7-14,\) find \(\mathbf{u} \cdot \mathbf{v}\) $$\mathbf{u}=3 \mathbf{i}+4 \mathbf{j}$$ $$\mathbf{v}=7 \mathbf{i}-2 \mathbf{j}$$

Trigonometric Form of a Complex Number \(\mathrm{In}\) Exercises \(11-30\) , represent the complex number graphically. Then write the trigonometric form of the number. $$5-5 i$$

In Exercises \(7-14,\) find \(\mathbf{u} \cdot \mathbf{v}\) $$\mathbf{u}=3 \mathbf{i}+2 \mathbf{j}$$ $$\mathbf{v}=-2 \mathbf{i}-3 \mathbf{j}$$

Trigonometric Form of a Complex Number \(\mathrm{In}\) Exercises \(11-30\) , represent the complex number graphically. Then write the trigonometric form of the number. $$1-\sqrt{3} i$$

Using the Law of Cosines, use the Law of Cosines to solve the triangle. Round your answers to two decimal places. $$ a=11, \quad b=15, \quad c=21 $$

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places. $$A=35^{\circ}, \quad B=65^{\circ}, \quad c=10$$

In Exercises \(7-14,\) find \(\mathbf{u} \cdot \mathbf{v}\) $$\mathbf{u}=\mathbf{i}-2 \mathbf{j}$$ $$\mathbf{v}=-2 \mathbf{i}+\mathbf{j}$$

Trigonometric Form of a Complex Number \(\mathrm{In}\) Exercises \(11-30\) , represent the complex number graphically. Then write the trigonometric form of the number. $$4-4 \sqrt{3} i$$

Using the Law of Cosines, use the Law of Cosines to solve the triangle. Round your answers to two decimal places. $$ a=55, \quad b=25, \quad c=72 $$

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places. $$A=120^{\circ}, \quad B=45^{\circ}, \quad c=16$$

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