Chapter 7

Fill in the blank. The ______ of a complex number \(a+b i\) is the distance between the origin \((0,0)\) and the point \((a, b).\)

Fill in the blank(s). The standard form of the Law of cosines for \(\cos C=\frac{a^{2}+b^{2}-c^{2}}{2 a b}\) is _____

Fill in the blank(s). A(n) _______ triangle has no right angles.

For two vectors \(\mathbf{u}\) and \(\mathbf{v},\) does \(\mathbf{u} \cdot \mathbf{v}=\mathbf{v} \cdot \mathbf{u} ?\)

Fill in the blank(s).A ___________ can be used to represent a quantity that involves both magnitude and directions.

Fill in the blank. ______ 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 blank(s). Law of sines: \(\frac{a}{\sin A}=\) ______________\(=\frac{c}{\sin C}\)

What is the dot product of two orthogonal vectors?

Fill in the blank(s).The directed line segment \(\overrightarrow{P Q}\) has _______ point \(P\) and ______ point \(Q .\)

Fill in the blank. The complex number \(u=a+b i\) is an _____ of the complex number \(z\) when \(z=u^{n}=(a+b i)^{n}\).

Is the dot product of two vectors an angle, a vector, or a scalar?

Fill in the blank(s).The ______ of the directed line segment \(\overrightarrow{P Q}\) is denoted by \(\|\overrightarrow{P Q}\|\).

What is the trigonometric form of the complex number \(z=a+b i ?\)

One of the cases for the known measures of an oblique triangle is given. State whether the Law of cosines can be used to solve the triangle. SAS

Fill in the blank(s). Two___________ and one_____________determine a unique triangle.

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

When a complex number is written in trigonometric form, what does \(r\) represent?

One of the cases for the known measures of an oblique triangle is given. State whether the Law of cosines can be used to solve the triangle. SSS

When a complex number is written in trigonometric form, what does \(\theta\) represent?

One of the cases for the known measures of an oblique triangle is given. State whether the Law of cosines can be used to solve the triangle. AAS

Is the longest side of an oblique triangle always opposite the largest angle of the triangle?

The work \(W\) done by a constant force \(\mathbf{F}\) as its point of application moves along the vector \(\overrightarrow{P Q}\) is given by either \(W=\)_____ or \(W=\)_____.

Fill in the blank(s).The two basic vector operations are scalar ______ and vector _____.

Plot the complex number and find its absolute value. $$9+7 i$$

Find the dot product of u and v. $$\begin{aligned} &\mathbf{u}=\langle 6,3\rangle\\\ &\mathbf{v}=\langle 2,-4\rangle \end{aligned}$$

Fill in the blank(s).The vector \(\mathbf{u}+\mathbf{v}\) is called the ______ of vector addition.

Plot the complex number and find its absolute value. $$10-3 i$$

Fill in the blank(s).The vector sum \(v_{1} \mathbf{i}+v_{2} \mathbf{j}\) is called a ______ of the vectors \(\mathbf{i}\) and \(\mathbf{j},\) and the scalars \(v_{1}\) and \(v_{2}\) are called the ______ and ______ components of \(\mathbf{v},\) respectively.

Find the dot product of u and v. $$\begin{aligned} &\mathbf{u}=\langle-4,1\rangle\\\ &\mathbf{v}=\langle 5,-4\rangle \end{aligned}$$

Plot the complex number and find its absolute value. $$-5-12 i$$

Fill in the blank(s).What two characteristics determine whether two directed line segments are equivalent?

Find the dot product of u and v. $$\begin{aligned} &\mathbf{u}=5 \mathbf{i}+\mathbf{j}\\\ &\mathbf{v}=3 \mathbf{i}-\mathbf{j} \end{aligned}$$

Plot the complex number and find its absolute value. $$-4+6 i$$

Fill in the blank(s).What do you call a vector that has a magnitude of \(1 ?\)

Find the dot product of u and v. $$\begin{aligned} &\mathbf{u}=2 \mathbf{i}+6 \mathbf{j}\\\ &\mathbf{v}=-3 \mathbf{i}+7 \mathbf{j} \end{aligned}$$

Use the vectors \(\mathbf{u}=\langle 2,2\rangle, \mathbf{v}=\langle-5,3\rangle,\) and \(\mathbf{w}=\langle\mathbf{1},-\mathbf{4}\rangle\) to find the indicated quantity. State whether the result is a vector or a scalar. $$\mathbf{u} \cdot \mathbf{u}$$

Plot the complex number and find its absolute value. $$9 i$$

Use the vectors \(\mathbf{u}=\langle 2,2\rangle, \mathbf{v}=\langle-5,3\rangle,\) and \(\mathbf{w}=\langle\mathbf{1},-\mathbf{4}\rangle\) to find the indicated quantity. State whether the result is a vector or a scalar. $$\mathbf{v} \cdot \mathbf{w}$$

Plot the complex number and find its absolute value. $$-2 i$$

Use the vectors \(\mathbf{u}=\langle 2,2\rangle, \mathbf{v}=\langle-5,3\rangle,\) and \(\mathbf{w}=\langle\mathbf{1},-\mathbf{4}\rangle\) to find the indicated quantity. State whether the result is a vector or a scalar. $$\mathbf{u} \cdot 2 \mathbf{v}$$

Use the Law of cosines to solve the triangle. $$a=11, \quad b=15, \quad c=21$$

Use the Law of sines to solve the triangle. \(A=36^{\circ}, \quad a=8, \quad b=5\)

Use the vectors \(\mathbf{u}=\langle 2,2\rangle, \mathbf{v}=\langle-5,3\rangle,\) and \(\mathbf{w}=\langle\mathbf{1},-\mathbf{4}\rangle\) to find the indicated quantity. State whether the result is a vector or a scalar. $$4\mathbf{u} \cdot \mathbf{v}$$

Use the Law of cosines to solve the triangle. $$a=9, \quad b=3, \quad c=11$$

Use the Law of sines to solve the triangle. \(A=76^{\circ}, \quad a=34, \quad b=21\)

Use the vectors \(\mathbf{u}=\langle 2,2\rangle, \mathbf{v}=\langle-5,3\rangle,\) and \(\mathbf{w}=\langle\mathbf{1},-\mathbf{4}\rangle\) to find the indicated quantity. State whether the result is a vector or a scalar. $$(3 \mathbf{w} \cdot \mathbf{v}) \mathbf{u}$$

Use the Law of cosines to solve the triangle. $$A=50^{\circ}, \quad b=15, \quad c=30$$

Use the Law of cosines to solve the triangle. $$C=108^{\circ}, \quad a=10, \quad b=7$$

Use the dot product to find the magnitude of \(\mathbf{u}.\) $$\mathbf{u}=\langle-5,12\rangle$$

Use the Law of cosines to solve the triangle. $$A=120^{\circ}, \quad b=6, \quad c=7$$