91Ó°ÊÓ

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

In Exercises \(17-24,\) sketch the graph of the equation in the complex plane (z denotes a complex number of the form a \(+b i\) ). \(|z|=4[\text {Hint}:\) The graph consists of all points that lie 4 units from the origin. \(]\)

Find the nth roots in polar form. $$16\left(\cos \frac{\pi}{7}+i \sin \frac{\pi}{7}\right) ; \quad n=5$$

Find the component form of the vector \(v\) whose magnitude and direction angle \(\theta\) are given. $$\|\mathbf{v}\|=4, \theta=0^{\circ}$$

Find the component form of the vector \(v\) whose magnitude and direction angle \(\theta\) are given. $$\|\mathbf{v}\|=6, \theta=40^{\circ}$$

Find the component form of the vector \(v\) whose magnitude and direction angle \(\theta\) are given. $$\|\mathbf{v}\|=3, \boldsymbol{\theta}=310^{\circ}$$

Find a unit vector that has the same direction as \(v\). $$5 \mathbf{i}+10 \mathbf{j}$$

Find a unit vector that has the same direction as \(v\). $$-3 \mathbf{i}-9 \mathbf{j}$$

An object at the origin is acted upon by two forces, \(u\) and \(v,\) with direction angle \(\theta_{u}\) and \(\theta_{w}\) respectively. Find the direction and magnitude of the resultant force. $$\mathbf{u}=30 \text { pounds, } \theta_{u}=0^{\circ} ; \mathbf{v}=90 \text { pounds, } \theta_{v}=60^{\circ}$$

Deal with an object on an inclined plane. The situation is similar to that in Figure \(9-20\) of Example \(12,\) where \(\|\overline{T P}\|\) is the component of the weight of the object parallel to the plane and \(\|\overline{T Q}\|\) is the component of the weight perpendicular to the plane. An object weighing 50 pounds lies on an inclined plane that makes a \(40^{\circ}\) angle with the horizontal. Find the components of the weight parallel and perpendicular to the plane.