2829 Sketch the solid in 3-space that is described in cylindricalcoordinates by the

(a) What is the difference between a vector and a scalar? Give a physical example of each. (b) How can you determine whether or not two vectors are orthogonal? (c) How can you determine whether or not two vectors are parallel? (d) How can you determine whether or not three vectors with a common initial point lie in the same plane in \(3\)-space? Equation Transcription: Text Transcription: 3

(a) Sketch vectors u and v for which \(u+v\) and \(u-v\) are orthogonal. (b) How can you use vectors to determine whether four points in 3-space lie in the same plane? (c) If forces \(F_{1}=i\) and \(F_{2}=j\) are applied at a point in 2 -space, what force would you apply at that point to cancel the combined effect of \(F_{1}\) and \(F_{2}\) ? (d) Write an equation of the sphere with center \((1,-2,2)\) that passes through the origin. Equation Transcription: Text Transcription: u+v u-v F_1 =i F_2 =j F_1 F_2 (1,-2,2)

(a) Draw a picture that shows the direction angles \(\alpha, \beta, \text { and } \gamma\) of a vector. (b) What are the components of a unit vector in 2 -space that makes an angle of \(120^{\circ}\) with the vector i (two answers)? (c) How can you use vectors to determine whether a triangle with known vertices \(P_{1}, P_{2}, \text { and } P_{3}\) has an obtuse angle? (d) True or false: The cross product of orthogonal unit vectors is a unit vector. Explain your reasoning. Equation Transcription: ° Text Transcription: alpha, beta, and gamma 120 degrees P_1, P_2, and P_3

(a) Make a table that shows all possible cross products of the vectors i, j, and k. (b) Give a geometric interpretation of \(\|u \times v\|\). (c) Give a geometric interpretation of \(|u \cdot(v \times w)|\). (d) Write an equation of the plane that passes through the origin and is perpendicular to the line \(x=t, y=2 t, z=-t\) Equation Transcription: Text Transcription: || u times v || | u . (v times w) | x=t, y=2t, z=-t

In each part, find an equation of the sphere with center \((-3,5,-4)\) and satisfying the given condition. (a) Tangent to the -plane (b) Tangent to the -plane (c) Tangent to the -plane Equation Transcription: Text Transcription: (-3, 5, -4)

Find the largest and smallest distances between the point \(P(1,1,1)\) and the sphere \(x^{2}+y^{2}+z^{2}-2 y+6 z-6=0\) Equation Transcription: Text Transcription: P(1, 1, 1) x^2 +y^2 +z^2 -2y+6z-6=0

Given the points \(P(3,4), Q(1,1), \text { and } R(5,2)\), use vector methods to find the coordinates of the fourth vertex of the parallelogram whose adjacent sides are and . Equation Transcription: Text Transcription: P(3, 4), Q(1, 1), and R(5, 2)

Let \(u=\langle 3,5,-1\rangle\) and \(v=\langle 2,-2,3\rangle\). Find (a) \(2 u+5 v\) (b) \(\frac{1}{\|v\|} v\) (c) \(\|u\|\) (d) \(\|u-v\|\) Equation Transcription: ?? ?? Text Transcription: u=langle 3, 5, -1 rangle v=langle 2, -2, 3 rangle 2u+5v 1/|| v || v ||u|| ||u-v||

Let \(a=c i+j\) and \(b=4 i+3 j\). Find c so that (a) a and b are orthogonal (b) the angle between a and b is \(\pi / 4\) (c) the angle between a and b is \(\pi / 6\) (d) a and b are parallel. Equation Transcription: Text Transcription: a=ci + j b=4i + 3j pi/4 pi/6

Let \(r_{0}=\left\langle x_{0}, y_{0}, z_{0}\right\rangle\) and \(r=\langle x, y, z\rangle\). Describe the set of all points \((x, y, z)\) for which (a) \(r \cdot r_{0}=0\) (b) \(\left(r-r_{0}\right) \cdot r_{0}=0\) Equation Transcription: ?? ?? Text Transcription: r_0 langle x_0, y_0, z_0 rangle r=langle x, y, z rangle (x, y, z) r . r_0 =0 (r-r_0).r_0 =0

Show that if u and v are unit vectors and \(\theta\) is the angle between them, then \(\|u-v\|=2 \sin \frac{1}{2} \theta\) Equation Transcription: Text Transcription: theta || u-v || = 2 sin 1/2 theta

Find the vector with length 5 and direction angles \(\alpha=60^{\circ}\), \(\beta=120^{\circ}\), \(\gamma=135^{\circ}\) Equation Transcription: ° ° ° Text Transcription: alpha = 60 degrees beta = 120 degrees gamma = 135 degrees

Assuming that force is in pounds and distance is in feet, find the work done by a constant force \(F=3 i-4 j+k\) acting on a particle that moves on a straight line from \(P(5,7,0)\) to \(Q(6,6,6)\). Equation Transcription: Text Transcription: F=3i-4j+k P(5, 7, 0) Q(6, 6, 6)

Assuming that force is in newtons and distance is in meters, find the work done by the resultant of the constant forces \(F_{1}=i-3 j+k\) and \(F_{2}=i+2 j+2 k\) acting on a particle that moves on a straight line from \(P(-1,-2,3)\) to \(Q(0,2,0)\) Equation Transcription: Text Transcription: F_1=i - 3j +k F_2 = i + 2j + 2k P(-1, -2, 3) Q(0, 2, 0)

(a) Find the area of the triangle with vertices \(A(1,0,1)\), \(B(0,2,3)\), and \(C(2,1,0)\) (b) Use the result in part (a) to find the length of the altitude from vertex to side . Equation Transcription: Text Transcription: A(1, 0, 1) B(0, 2, 3) C(2, 1, 0)

True or false? Explain your reasoning. (a) If \(u \cdot v=0\), then \(u=0\) or \(v=0\) (b) If \(u \times v=0\), then \(u=0\) or \(v=0\). (c) If \(u \cdot v=0\) and \(u \times v=0\), then \(u=0\) or \(v=0\). Equation Transcription: Text Transcription: u . v =0 u=0 v=0 u times v =0 u=0 v=0 u . v =0 u times v =0 u=0 v=0

Consider the points \(A(1,-1,2), B(2,-3,0), C(-1,-2,0), D(2,1,-1)\) (a) Find the volume of the parallelepiped that has the vectors as adjacent edges. (b) Find the distance from D to the plane containing A, B, and C. Equation Transcription: Text Transcription: A(1, -1, 2), B(2, -3, 0), C(-1, -2, 0), D(2, 1, -1)

Suppose that a force \(F\) with a magnitude of 9lb is applied to the lever-shaft assembly shown in the accompanying figure. (a) Express the force \(F\) in component form. (b) Find the vector moment of \(F\) about the origin. Equation Transcription: Text Transcription: F

Let P be the point \((4,1,2)\). Find parametric equations for the line through P and parallel to the vector \(\{1,-1.0\}\). Equation Transcription: Text Transcription: (4, 1, 2) {1, -1, 0}

(a) Find parametric equations for the intersection of the planes \(2 x+y-z=3\) and \(x+2 y+z=3\) (b) Find the acute angle between the two planes. Equation Transcription: Text Transcription: 2x+y-z=3 x+2y+z=3

Find an equation of the plane that is parallel to the plane \(x+5 y-z+8=0\) and contains the point \((1,1,4)\) Equation Transcription: Text Transcription: x+5y-z+8=0 (1, 1, 4)

Find an equation of the plane through the point \((4,3,0)\) and parallel to the vectors \(i+k\) and \(2 j-k\). Equation Transcription: Text Transcription: (4, 3, 0) i+k 2j-k

What condition must the constants satisfy for the planes \(a_{1} x+b_{1} y+c_{1} z=d_{1}\) and \(a_{2} x+b_{2} y+c_{2} z=d_{2}\) to be perpendicular? Equation Transcription: Text Transcription: a_1 x+b_1 y+c_1 z=d_1 a_2 x +b_2 y+c_2 z=d_2

(a) List six common types of quadric surfaces, and describe their traces in planes parallel to the coordinate planes. (b) Give the coordinates of the points that result when the point \((x, y, z)\) is reflected about the plane \(y=x\), the plane \(y=z\), and the plane \(x=z\) (c) Describe the intersection of the surfaces \(r=5\) and \(z=1\) in cylindrical coordinates. (d) Describe the intersection of the surfaces \(\varphi=\pi / 4\) and \(\theta=0\) in spherical coordinates. Equation Transcription: Text Transcription: (x, y, z) y=x y=z x=z r=5 z=1 phi=pi/4 theta=0

In each part, identify the surface by completing the squares. (a) \(x^{2}+4 y^{2}-z^{2}-6 x+8 y+4 z=0\) (b) \(x^{2}+y^{2}+z^{2}+6 x-4 y+12 z=0\) (c) \(x^{2}+y^{2}-z^{2}-2 x+4 y+5=0\) Equation Transcription: Text Transcription: x^2 +4y^2 -z^2 -6x +8y +4z =0 x^2 +y^2 +z^2 +6x -4y +12z =0 x^2 +y^2 -z^2 -2x +4y+5=0

In each part, express the equation in cylindrical and spherical coordinates. (a) \(x^{2}+y^{2}=z\) (b) \(x^{2}-y^{2}-z^{2}=0\) Equation Transcription: Text Transcription: x^2 +y^2 =z x^2 -y^2 -z^2 =0

In each part, express the equation in rectangular coordinates. (a) \(z=r^{2} \cos 2 \theta\) (b) \(\rho^{2} \sin \varphi \cos \varphi \cos \theta=1\) Equation Transcription: Text Transcription: z=r^2 cos 2 theta rho^2 sin phi cos phi cos theta =1

Sketch the solid in 3 -space that is described in cylindrical coordinates by the stated inequalities. (a) \(1 \leq r \leq 2\) (b) \(2 \leq z \leq 3\) (c) \(\pi / 6 \leq \theta \leq \pi / 3\) (d) \(1 \leq r \leq 2,2 \leq z \leq 3\), and \(\pi / 6 \leq \theta \leq \pi / 3\) Equation Transcription: Text Transcription: 1 leq r leq 2 2 leq z leq 3 pi/6 leq theta leq pi/3 1 leq r leq 2, 2 leq z leq 3 pi/6 leq theta leq pi/3

Sketch the solid in 3 -space that is described in cylindrical coordinates by the stated inequalities. (a) \(r^{2}+z^{2} \leq 4\) (b) \(r \leq 1\) (c) \(r^{2}+z^{2} \leq 4 \text { and } r>1\) Equation Transcription: Text Transcription: r^2 + z^2 leq 4 r leq 1 r^2 + z^2 leq 4 and r > 1

Sketch the solid in 3-space that is described in spherical coordinates by the stated inequalities. (a) \(0 \leq \rho \leq 2\) (b) \(0 \leq \varphi \leq \pi / 6\) (c) \(0 \leq \rho \leq 2 \text { and } 0 \leq \varphi \leq \pi / 6\) Equation Transcription: Text Transcription: 0 leq rho leq 2 0 leq phi leq pi/6 0 leq rho leq 2 and 0 leq phi leq pi/6

Sketch the solid in 3-space that is described in spherical coordinates by the stated inequalities. (a) \(0 \leq \rho \leq 5,0 \leq \varphi \leq \pi / 2, \text { and } 0 \leq \theta \leq \pi / 2\) (b) \(0 \leq \varphi \leq \pi / 3 \text { and } 0 \leq \rho \leq 2 \sec \varphi\) (c) \(0 \leq \rho \leq 2 \text { and } \pi / 6 \leq \varphi \leq \pi / 3\) Equation Transcription: Text Transcription: 0 leq rho leq 5, 0 leq phi leq pi/2, and 0 leq theta leq pi/2 0 leq phi leq pi/3 and 0 leq rho leq 2 sec phi 0 leq rho leq 2 and pi/6 leq phi leq pi/3

Sketch the surface whose equation in spherical coordinates is \(\rho=a(1-\cos \varphi)\). [Hint: The surface is shaped like a familiar fruit.] Equation Transcription: Text Transcription: rho=a(1-cos phi)