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Chapter 6: Problem 40

In Exercises \(31-40,\) find the angle \(\theta\) between the vectors. $$ \begin{array}{l}{\mathbf{u}=\cos \left(\frac{\pi}{4}\right) \mathbf{i}+\sin \left(\frac{\pi}{4}\right) \mathbf{j}} \\ {\mathbf{v}=\cos \left(\frac{\pi}{2}\right) \mathbf{i}+\sin \left(\frac{\pi}{2}\right) \mathbf{j}}\end{array} $$

Short Answer

Expert verified
\(\theta = \dfrac{\pi}{2}\) or 90 degrees.

Step by step solution

01

Calculate dot product of vectors

The dot product of two vectors is given as \(\mathbf{u} \cdot \mathbf{v} = u_i v_i + u_j v_j\). Substituting the given vectors \(\mathbf{u} = \cos\left(\dfrac{\pi}{4} \right) \mathbf{i} + \sin\left(\dfrac{\pi}{4} \right) \mathbf{j}\) and \(\mathbf{v} = \cos\left (\dfrac{\pi}{2} \right) \mathbf{i} + \sin\left(\dfrac{\pi}{2} \right) \mathbf{j}\) into the formula, we get the dot product as \(u \cdot v = 0\). This is because \(cos(\dfrac{\pi}{2}) = 0\) and \(sin(\dfrac{\pi}{4})\) times \(sin(\dfrac{\pi}{2}) = \dfrac{1}{\sqrt{2}}\).
02

Calculate magnitudes of vectors

The magnitude of a vector is given by \(\sqrt{u_i^2 + u_j^2}\). For \(\mathbf{u}\), \(|\mathbf{u}| = \sqrt{{\cos^2\left(\dfrac{\pi}{4} \right)+ \sin^2\left(\dfrac{\pi}{4} \right)}} = 1\) because \(cos(\dfrac{\pi}{4}) = sin(\dfrac{\pi}{4}) = \dfrac{1}{\sqrt{2}}\), and the square of these values added together equals 1. Similarly, for \(\mathbf{v}\), \( |\mathbf{v}| = \sqrt{{\cos^2\left(\dfrac{\pi}{2} \right)+ \sin^2\left(\dfrac{\pi}{2} \right)}} = 1\).
03

Calculate the angle

The angle between two vectors is given by \(\cos\theta = \dfrac{\mathbf{u} \cdot \mathbf{v}} {|\mathbf{u}|*|\mathbf{v}|}\). Plug all calculated values into this equation, \(\cos\theta = \dfrac{0} {1*1} = 0\). Therefore, the angle \(\theta = \cos^{-1}(0) = \dfrac{\pi}{2}\) or 90 degrees.

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Most popular questions from this chapter

In Exercises \(47-58\) , perform the operation and leave the result in trigonometric form. $$ \left[\frac{3}{4}\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)\right]\left[4\left(\cos \frac{3 \pi}{4}+i \sin \frac{3 \pi}{4}\right)\right] $$

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

Given two complex numbers \(z_1 = r_1(\cos\ \theta_1 + i\ \sin\ \theta_1)\) and \(z_2 = r_2(\cos\ \theta_2 + i\ \sin\ \theta_2)\), \(z_2 \neq 0\), show that \(\frac{z_1}{z_2} = \frac{r_1}{r_2}[\cos(\theta_1 - \theta_2) +\ i\ \sin(\theta_1 - \theta_2)]\).

In Exercises 47-58, perform the operation and leave the result in trigonometric form. \([\frac{5}{3}(\cos\ 120^{\circ} + i\ \sin\ 120^{\circ})][\frac{2}{3}(\cos\ 30^{\circ} + i\ \sin\ 30^{\circ})]\)

TRUE OR FALSE? In Exercises 107 and 108, determine whether the statement is true or false. Justify your answer. The product of two complex numbers is zero only when the modulus of one (or both) of the complex numbers is zero.
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