Chapter 9

Find \(\boldsymbol{u} \cdot \boldsymbol{v}, \boldsymbol{u} \cdot \boldsymbol{u},\) and \(\boldsymbol{v} \cdot \boldsymbol{v}\) $$\mathbf{u}=\langle 3,4\rangle, \mathbf{v}=\langle-5,2\rangle$$

Find the magnitude of the vector \(\overrightarrow{P Q}\). $$P=(2,3), Q=(5,9)$$

In Exercises \(1-8,\) plot the point in the complex plane corresponding to the number. $$3+2 i$$

Calculate the given product and express your answer in the form \(a+b i\). $$\left(\cos \frac{\pi}{12}+i \sin \frac{\pi}{12}\right)^{6}$$

Find \(\boldsymbol{u} \cdot \boldsymbol{v}, \boldsymbol{u} \cdot \boldsymbol{u},\) and \(\boldsymbol{v} \cdot \boldsymbol{v}\) $$\mathbf{u}=\langle-1,6\rangle, \mathbf{v}=\langle-4,1 / 3\rangle$$

Find the magnitude of the vector \(\overrightarrow{P Q}\). $$P=(-3,5), Q=(7,-11)$$

In Exercises \(1-8,\) plot the point in the complex plane corresponding to the number. $$-7+6 i$$

Calculate the given product and express your answer in the form \(a+b i\). $$\left(\cos \frac{\pi}{5}+i \sin \frac{\pi}{5}\right)^{20}$$

Find \(\boldsymbol{u} \cdot \boldsymbol{v}, \boldsymbol{u} \cdot \boldsymbol{u},\) and \(\boldsymbol{v} \cdot \boldsymbol{v}\) $$\mathbf{u}=2 \mathbf{i}+\mathbf{j}, \mathbf{v}=3 \mathbf{i}$$

Find the magnitude of the vector \(\overrightarrow{P Q}\). $$P=(-7,0), Q=(-4,-5)$$

In Exercises \(1-8,\) plot the point in the complex plane corresponding to the number. $$-\frac{8}{3}-\frac{5}{3} i$$

Calculate the given product and express your answer in the form \(a+b i\). $$\left[2\left(\cos \frac{\pi}{24}+i \sin \frac{\pi}{24}\right)\right]^{8}$$

Find \(\boldsymbol{u} \cdot \boldsymbol{v}, \boldsymbol{u} \cdot \boldsymbol{u},\) and \(\boldsymbol{v} \cdot \boldsymbol{v}\) $$\mathbf{u}=\mathbf{i}-\mathbf{j}, \mathbf{v}=5 \mathbf{j}$$

Find the magnitude of the vector \(\overrightarrow{P Q}\). $$P=(30,12), Q=(25,5)$$

In Exercises \(1-8,\) plot the point in the complex plane corresponding to the number. $$\sqrt{2}-7 i$$

Find \(\boldsymbol{u} \cdot \boldsymbol{v}, \boldsymbol{u} \cdot \boldsymbol{u},\) and \(\boldsymbol{v} \cdot \boldsymbol{v}\) $$\mathbf{u}=3 \mathbf{i}+2 \mathbf{j}, \mathbf{v}=2 \mathbf{i}+3 \mathbf{j}$$

Find a vector with the origin as initial point that is equivalent to the vector \(\overrightarrow{P Q}\). $$P=(1,5), Q=(7,11)$$

In Exercises \(1-8,\) plot the point in the complex plane corresponding to the number. $$(1+i)(1-i)$$

Calculate the given product and express your answer in the form \(a+b i\). $$\left[3\left(\cos \frac{7 \pi}{30}+i \sin \frac{7 \pi}{30}\right)\right]^{5}$$

Find \(\boldsymbol{u} \cdot \boldsymbol{v}, \boldsymbol{u} \cdot \boldsymbol{u},\) and \(\boldsymbol{v} \cdot \boldsymbol{v}\) $$\mathbf{u}=4 \mathbf{i}-\mathbf{j}, \mathbf{v}=-\mathbf{i}+2 \mathbf{j}$$

Find a vector with the origin as initial point that is equivalent to the vector \(\overrightarrow{P Q}\). $$P=(2,7), Q=(-2,9)$$

In Exercises \(1-8,\) plot the point in the complex plane corresponding to the number. $$(2+i)(1-2 i)$$

Calculate the given product and express your answer in the form \(a+b i\). $$\left[\sqrt[3]{4}\left(\cos \frac{7 \pi}{36}+i \sin \frac{7 \pi}{36}\right)\right]^{12}$$

Find the dot product when \(u=\langle 4,3\rangle\) \(\boldsymbol{v}=\langle-5,2\rangle,\) and \(\boldsymbol{w}=\langle 4,-1\rangle\) $$\mathbf{u} \cdot(\mathbf{v}+\mathbf{w})$$

Find a vector with the origin as initial point that is equivalent to the vector \(\overrightarrow{P Q}\). $$P=(-4,-8), Q=(-10,2)$$

Calculate the product by expressing the number in polar form and using DeMoivre's Theorem. Express your answer in the form \(a+b i\). $$\left(\frac{1}{2}+\frac{\sqrt{3}}{2} i\right)^{3}$$

In Exercises \(1-8,\) plot the point in the complex plane corresponding to the number. $$2 i\left(3-\frac{5}{2} i\right)$$

Find the dot product when \(u=\langle 4,3\rangle\) \(\boldsymbol{v}=\langle-5,2\rangle,\) and \(\boldsymbol{w}=\langle 4,-1\rangle\) $$\mathbf{u}^{*}(\mathbf{v}-\mathbf{w})$$

Find a vector with the origin as initial point that is equivalent to the vector \(\overrightarrow{P Q}\). $$P=(-5,6), Q=(-7,-9)$$

Calculate the product by expressing the number in polar form and using DeMoivre's Theorem. Express your answer in the form \(a+b i\). $$\left(-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} i\right)^{4}$$

In Exercises \(1-8,\) plot the point in the complex plane corresponding to the number. $$\frac{4 i}{3}(-6-3 i)$$

Find the dot product when \(u=\langle 4,3\rangle\) \(\boldsymbol{v}=\langle-5,2\rangle,\) and \(\boldsymbol{w}=\langle 4,-1\rangle\) $$(\mathbf{u}+\mathbf{v}) \cdot(\mathbf{v}+\mathbf{w})$$

Find a vector with the origin as initial point that is equivalent to the vector \(\overrightarrow{P Q}\). $$P=\left(\frac{4}{5},-2\right), Q=\left(\frac{17}{5},-\frac{12}{5}\right)$$

Calculate the product by expressing the number in polar form and using DeMoivre's Theorem. Express your answer in the form \(a+b i\). $$(1-i)^{12}$$

In Exercises \(9-14,\) find the absolute value. $$|5-12 i|$$

Find a vector with the origin as initial point that is equivalent to the vector \(\overrightarrow{P Q}\). $$P=(\sqrt{2}, 4), Q=(\sqrt{3},-1)$$

Calculate the product by expressing the number in polar form and using DeMoivre's Theorem. Express your answer in the form \(a+b i\). $$(2+2 i)^{8}$$

In Exercises \(9-14,\) find the absolute value. $$|2 i|$$

Find \(u+v, v-u,\) and \(2 u-3 v\). $$\mathbf{u}=\langle-2,4\rangle, \mathbf{v}=\langle 6,1\rangle$$

Calculate the product by expressing the number in polar form and using DeMoivre's Theorem. Express your answer in the form \(a+b i\). $$\left(\frac{\sqrt{3}}{2}+\frac{1}{2} i\right)^{10}$$

In Exercises \(9-14,\) find the absolute value. $$|1+\sqrt{2} i|$$

Find \(u+v, v-u,\) and \(2 u-3 v\). $$\mathbf{u}=\langle 4,0\rangle, \mathbf{v}=\langle 1,-3\rangle$$

Calculate the product by expressing the number in polar form and using DeMoivre's Theorem. Express your answer in the form \(a+b i\). $$\left(-\frac{1}{2}+\frac{\sqrt{3}}{2} i\right)^{20}$$

In Exercises \(9-14,\) find the absolute value. $$|2-3 i|$$

Find the angle between the two vectors. $$\langle 4,-3\rangle,\langle 1,2\rangle$$

Find \(u+v, v-u,\) and \(2 u-3 v\). $$\mathbf{u}=\langle 3,3 \sqrt{2}\rangle, \mathbf{v}=\langle 4 \sqrt{2}, 1\rangle$$

Calculate the product by expressing the number in polar form and using DeMoivre's Theorem. Express your answer in the form \(a+b i\). $$\left(\frac{-1}{\sqrt{2}}+\frac{i}{\sqrt{2}}\right)^{14}$$

In Exercises \(9-14,\) find the absolute value. $$|-12 i|$$

Find the angle between the two vectors. $$\langle 2,4\rangle,\langle 0,-5\rangle$$

Find \(u+v, v-u,\) and \(2 u-3 v\). $$\mathbf{u}=\left\langle\frac{2}{3}, 4\right\rangle, \mathbf{v}=\left\langle-7, \frac{19}{3}\right\rangle$$