91Ó°ÊÓ

For the following exercises, the rectangular coordinates \((x, y, z)\) of a point are given. Find the spherical coordinates \((\rho, \theta, \varphi)\) of the point. Express the measure of the angles in degrees rounded to the nearest integer.\((0,3,0)\)

For the following exercises, point \(P\) and vector \(\mathbf{n}\) are given. Find the scalar equation of the plane that passes through \(P\) and has normal vector \(\mathbf{n}\). Find the general form of the equation of the plane that passes through \(P\) and has normal vector \(\mathbf{n}\). \(P(0,0,0), \mathbf{n}=3 \mathbf{i}-2 \mathbf{j}+4 \mathbf{k}\)

Consider \(\mathbf{u}\) and \(\mathbf{v}\) two three-dimensional vectors. If the magnitude of the cross product vector \(\mathbf{u} \times \mathbf{v}\) is \(k\) times larger than the magnitude of vector \(\mathbf{u}\), show that the magnitude of \(\mathbf{v}\) is greater than or equal to \(k\), where \(k\) is a natural number.

Consider the parabolic reflector described by equation \(z=20 x^{2}+20 y^{2} .\) Find its focal point.

For the following exercises, point \(P\) and vector \(\mathbf{n}\) are given. Find the scalar equation of the plane that passes through \(P\) and has normal vector \(\mathbf{n}\). Find the general form of the equation of the plane that passes through \(P\) and has normal vector \(\mathbf{n}\). \(P(1,2,3), \mathbf{n}=\langle 1,2,3\rangle\)

Find the work done by force \(\mathbf{F}=\langle 5,6,-2\rangle\) (measured in Newtons) that moves a particle from point \(P(3,-1,0)\) to point \(Q(2,3,1)\) along a straight line (the distance is measured in meters).

The intersection between cylinder \((x-1)^{2}+y^{2}=1\) and sphere \(x^{2}+y^{2}+z^{2}=4\) is called a Viviani curve. a. Solve the system consisting of the equations of the surfaces to find the equation of the intersection curve. (Hint: Find \(x\) and \(y\) in terms of \(z .)\) b. Use a computer algebra system (CAS) to visualize the intersection curve on sphere \(x^{2}+y^{2}+z^{2}=4\).

For the following exercises, the equation of a plane is given. Find normal vector \(\mathbf{n}\) to the plane. Express \(\mathbf{n}\) using standard unit vectors. Find the intersections of the plane with the axes of coordinates. Sketch the plane. \(x+z=0\)

What is the magnitude of the force required to be applied to the end of a 1-ft wrench at an angle of \(35^{\circ}\) to produce a torque of \(20 \mathrm{~N} \cdot \mathrm{m} ?\)

Two soccer players are practicing for an upcoming game. One of them runs \(10 \mathrm{~m}\) from point \(A\) to point \(B\). She then turns left at \(90^{\circ}\) and runs \(10 \mathrm{~m}\) until she reaches point \(C\). Then she kicks the ball with a speed of \(10 \mathrm{~m} / \mathrm{sec}\) at an upward angle of \(45^{\circ}\) to her teammate, who is located at point \(A\). Write the velocity of the ball in component form.