Calculus Volume 3
16th Edition
ISBN: 9781938168079
Author: Gilbert Strang, Edwin Jed Herman
Publisher: OpenStax
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Textbook Question
Chapter 2.2, Problem 91E
For the following exercises, find the unit vector in the direction of the given vector
91.
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1.2.19. Let and s be natural numbers. Let G be the simple graph with vertex set
Vo... V„−1 such that v; ↔ v; if and only if |ji| Є (r,s). Prove that S has exactly k
components, where k is the greatest common divisor of {n, r,s}.
Question 3
over a field K.
In this question, MË(K) denotes the set of n × n matrices
(a) Suppose that A Є Mn(K) is an invertible matrix. Is it always true that A is
equivalent to A-¹? Justify your answer.
(b) Let B be given by
8
B = 0 7 7
0 -7 7
Working over the field F2 with 2 elements, compute the rank of B as an element
of M2(F2).
(c) Let
1
C
-1 1
[4]
[6]
and consider C as an element of M3(Q). Determine the minimal polynomial
mc(x) and hence, or otherwise, show that C can not be diagonalised.
[7]
(d) Show that C in (c) considered as an element of M3(R) can be diagonalised. Write
down all the eigenvalues. Show your working.
[8]
16. Solve the given differential equation:
y" + 4y sin (t)u(t 2π),
-
y(0) = 1, y'(0) = 0
Given,
1
(x² + 1)(x²+4)
1/3
-1/3
=
+
x²+1 x² +4
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Chapter 2 Solutions
Calculus Volume 3
Ch. 2.1 - For the following exercises, consider points...Ch. 2.1 - For the following exercises, consider points...Ch. 2.1 - For the following exercises, consider points...Ch. 2.1 - For the following exercises, consider points...Ch. 2.1 - For the following exercises, consider points...Ch. 2.1 - For the following exercises, consider points...Ch. 2.1 - For the following exercises, consider points...Ch. 2.1 - For the following exercises, consider points...Ch. 2.1 - For the following exercises, consider points...Ch. 2.1 - For the following exercises, consider points...
Ch. 2.1 - For the following exercises, consider points...Ch. 2.1 - For the following exercises, consider points...Ch. 2.1 - For the following exercises, consider points...Ch. 2.1 - For the following exercises, consider points...Ch. 2.1 - Far the following exercises, use the given...Ch. 2.1 - Far the following exercises, use the given...Ch. 2.1 - Far the following exercises, use the given...Ch. 2.1 - Far the following exercises, use the given...Ch. 2.1 - Far the following exercises, use the given...Ch. 2.1 - Far the following exercises, use the given...Ch. 2.1 - Far the following exercises, use the given...Ch. 2.1 - Far the following exercises, use the given...Ch. 2.1 - Far the following exercises, use the given...Ch. 2.1 - Far the following exercises, use the given...Ch. 2.1 - For the following exercises, find vector v with...Ch. 2.1 - For the following exercises, find vector v with...Ch. 2.1 - For the following exercises, find vector v with...Ch. 2.1 - For the following exercises, find vector v with...Ch. 2.1 - For the following exercises, find the component...Ch. 2.1 - For the following exercises, find the component...Ch. 2.1 - For the following exercises, find the component...Ch. 2.1 - For the following exercises, find the component...Ch. 2.1 - For the following exercises, find the component...Ch. 2.1 - For the following exercises, find the component...Ch. 2.1 - For the following exercises, vector u is given....Ch. 2.1 - For the following exercises, vector u is given....Ch. 2.1 - Let a=a1,a2,b=b1,b2 , and c=c1,c2 be three nonzero...Ch. 2.1 - Consider vectors a=2,4,b=1,2, and c=0 . Determine...Ch. 2.1 - Let P(x0,f(x0)) be a fixed point on the graph of...Ch. 2.1 - Consider the function f(x)=x4 , where x. Determine...Ch. 2.1 - Consider f and g two functions defined on the same...Ch. 2.1 - Find x such that vectors a=x,sinx and b=x,cosx are...Ch. 2.1 - Calculate the coordinates of point D such that...Ch. 2.1 - Consider the points A(2,1),B(10,6),C(13,4) , and...Ch. 2.1 - The speed of an object is the magnitude of its...Ch. 2.1 - A baseball player throws a baseball at an angle of...Ch. 2.1 - A bullet is fired with an initial velocity of...Ch. 2.1 - [T] A 65kg sprinter exerts a force of 798N at a 19...Ch. 2.1 - [T] Two forces, a horizontal force of 45lb and...Ch. 2.1 - [T] Two fumes, a vertical fence of 26lb and...Ch. 2.1 - [T] Three forces act an object. Two of the fences...Ch. 2.1 - Three forces with magnitudes 80lb,120lb, and 60lb...Ch. 2.1 - [T] An airplane is flying in the direction of 43...Ch. 2.1 - [T] A boat is traveling in the water at 30mph in a...Ch. 2.1 - [T] A 50-lb weight is hung by a cable so that the...Ch. 2.1 - [T] A 62-lh weight hangs from a nope that makes...Ch. 2.1 - [T] A 1500lb boat is parked on a ramp that makes...Ch. 2.1 - [T] An 85lb box is at rest on a 26 incline....Ch. 2.1 - A guy-wire supports a pole that is 75ft high. One...Ch. 2.1 - A telephone pole guy-wire has an angle of...Ch. 2.2 - Consider a rectangular box with one of the...Ch. 2.2 - Find the coordinates of point P and determine its...Ch. 2.2 - For the following exercises, describe and graph...Ch. 2.2 - For the following exercises, describe and graph...Ch. 2.2 - For the following exercises, describe and graph...Ch. 2.2 - For the following exercises, describe and graph...Ch. 2.2 - Write the equation of the plane passing through...Ch. 2.2 - Write the equation of the plane passing through...Ch. 2.2 - Find an equation of the plane passing through...Ch. 2.2 - Find an equation of the plane passing through...Ch. 2.2 - For the following exercises, find the equation of...Ch. 2.2 - For the following exercises, find the equation of...Ch. 2.2 - For the following exercises, find the equation of...Ch. 2.2 - For the following exercises, find the equation of...Ch. 2.2 - For the following exercises, find the center and...Ch. 2.2 - For the following exercises, find the center and...Ch. 2.2 - For the following exercises, express vector PQ...Ch. 2.2 - For the following exercises, express vector PQ...Ch. 2.2 - For the following exercises, express vector PQ...Ch. 2.2 - For the following exercises, express vector PQ...Ch. 2.2 - For the following exercises, express vector PQ...Ch. 2.2 - For the following exercises, express vector PQ...Ch. 2.2 - For the following exercises, use the given vectors...Ch. 2.2 - For the following exercises, use the given vectors...Ch. 2.2 - For the following exercises, use the given vectors...Ch. 2.2 - For the following exercises, use the given vectors...Ch. 2.2 - For the following exercises, vectors u and v are...Ch. 2.2 - For the following exercises, vectors u and v are...Ch. 2.2 - For the following exercises, vectors u and v are...Ch. 2.2 - For the following exercises, vectors u and v are...Ch. 2.2 - For the following exercises, find the unit vector...Ch. 2.2 - For the following exercises, find the unit vector...Ch. 2.2 - For the following exercises, find the unit vector...Ch. 2.2 - For the following exercises, find the unit vector...Ch. 2.2 - For the following exercises, find the unit vector...Ch. 2.2 - For the following exercises, find the unit vector...Ch. 2.2 - Determine whether AB and PQ are equivalent...Ch. 2.2 - Determine whether the vectors AB and PQ are...Ch. 2.2 - For the following exercises, find vector u with a...Ch. 2.2 - For the following exercises, find vector u with a...Ch. 2.2 - For the following exercises, find vector u with a...Ch. 2.2 - For the following exercises, find vector u with a...Ch. 2.2 - Determine a vector of magnitude 5 in the direction...Ch. 2.2 - Find a vector of magnitude 2 that points in the...Ch. 2.2 - Consider the points A(2,,0),B(0,1,), and C(1,1,),...Ch. 2.2 - Consider points A(,0,0),B(0,,0), and C(,,), where ...Ch. 2.2 - Let P(x,y,z) be a point situated at an equal...Ch. 2.2 - Let P(x,y,z) be a point situated at an equal...Ch. 2.2 - The points A,B, and C are collinear (in this...Ch. 2.2 - Show that points A(1,0,1),B(0,1,1) and C(1,1,1)...Ch. 2.2 - [T] A force F of 50N acts on a particle in the...Ch. 2.2 - [T] A force F of 40N acts on a box in the...Ch. 2.2 - If F is a force that moves an object from point...Ch. 2.2 - A box is pulled 4 yd horizontally in the x...Ch. 2.2 - The sum of the forces acting on an object is...Ch. 2.2 - [T] Let Fk=1,k,k2,k=1,...,n be n forces acting on...Ch. 2.2 - The force of gravity F acting on an object is...Ch. 2.2 - A 5-kg pendant chandelier is designed such that...Ch. 2.2 - [T] A 30-kg block of cement is suspended by three...Ch. 2.2 - Two soccer players are practicing for an upcoming...Ch. 2.2 - Let r(t)=x(t),y(t),z(t) be the position vector of...Ch. 2.2 - [T] Let r(t)=t,2t2,4t2 be the position vector of a...Ch. 2.3 - For the following exercises, the vectors u and v...Ch. 2.3 - For the following exercises, the vectors u and v...Ch. 2.3 - For the following exercises, the vectors u and v...Ch. 2.3 - For the following exercises, the vectors u and v...Ch. 2.3 - For the following exercises, the vectors a,b, and...Ch. 2.3 - For the following exercises, the vectors a,b, and...Ch. 2.3 - For the following exercises, the vectors a,b, and...Ch. 2.3 - For the following exercises, the vectors a,b, and...Ch. 2.3 - For the following exercises, the two-dimensional...Ch. 2.3 - For the following exercises, the two-dimensional...Ch. 2.3 - For the following exercises, the two-dimensional...Ch. 2.3 - For the following exercises, the two-dimensional...Ch. 2.3 - For the following exercises, find the measure 0f...Ch. 2.3 - For the following exercises, find the measure 0f...Ch. 2.3 - For the following exercises, find the measure 0f...Ch. 2.3 - For the following exercises, find the measure 0f...Ch. 2.3 - For the following exercises, find the measure 0f...Ch. 2.3 - For the following exercises, find the measure 0f...Ch. 2.3 - For the following exercises determine whether the...Ch. 2.3 - For the following exercises determine whether the...Ch. 2.3 - For the following exercises determine whether the...Ch. 2.3 - For the following exercises determine whether the...Ch. 2.3 - For the following exercises determine whether the...Ch. 2.3 - For the following exercises determine whether the...Ch. 2.3 - For the following exercises determine whether the...Ch. 2.3 - For the following exercises determine whether the...Ch. 2.3 - For the following exercises determine whether the...Ch. 2.3 - For the following exercises determine whether the...Ch. 2.3 - [T] Consider the points P(4,5) and Q(5,7) ....Ch. 2.3 - [T] Consider points A(1,1),B(2,7) , and C(6,3)....Ch. 2.3 - Determine the measure of angle A in triangle ABC ,...Ch. 2.3 - Consider points P(3,7,2) and Q(1,1,3). Determine...Ch. 2.3 - For the following exercises, determine which (if...Ch. 2.3 - For the following exercises, determine which (if...Ch. 2.3 - For the following exercises, determine which (if...Ch. 2.3 - For the following exercises, determine which (if...Ch. 2.3 - For the following exercises, determine which (if...Ch. 2.3 - For the following exercises, determine which (if...Ch. 2.3 - For the following problems, the vector u is given....Ch. 2.3 - For the following problems, the vector u is given....Ch. 2.3 - For the following problems, the vector u is given....Ch. 2.3 - For the following problems, the vector u is given....Ch. 2.3 - For the following problems, the vector u is given....Ch. 2.3 - For the following problems, the vector u is given....Ch. 2.3 - For the following exercises, the vectors u and v...Ch. 2.3 - For the following exercises, the vectors u and v...Ch. 2.3 - For the following exercises, the vectors u and v...Ch. 2.3 - For the following exercises, the vectors u and v...Ch. 2.3 - For the following exercises, the vectors u and v...Ch. 2.3 - For the following exercises, the vectors u and v...Ch. 2.3 - A methane molecule has a carbon atom situated at...Ch. 2.3 - [T] Find the vectors that join the center of a...Ch. 2.3 - Find the work done by force F=5,6,2 (measured in...Ch. 2.3 - [T] A sled is pulled by exerting a force of 100N...Ch. 2.3 - [T] A father is pulling his son on a sled at an...Ch. 2.3 - [T] A car is towed using a force of 1600N . The...Ch. 2.3 - [T] A boat sails north aided by a wind blowing in...Ch. 2.3 - Vector p=150,225,375 represents the price of...Ch. 2.3 - [T] Two forces F1 and F2 are represented by...Ch. 2.3 - [T] Consider r(t)=cost,sint,2t the position vector...Ch. 2.4 - For the following exercises, the vectors u and v...Ch. 2.4 - For the following exercises, the vectors u and v...Ch. 2.4 - For the following exercises, the vectors u and v...Ch. 2.4 - For the following exercises, the vectors u and v...Ch. 2.4 - `For the following exercises, the vectors u and v...Ch. 2.4 - For the following exercises, the vectors u and v...Ch. 2.4 - In the following exercises, vectors u and v are...Ch. 2.4 - In the following exercises, vectors u and v are...Ch. 2.4 - In the following exercises, vectors u and v are...Ch. 2.4 - In the following exercises, vectors u and v are...Ch. 2.4 - In the following exercises, vectors u and v are...Ch. 2.4 - In the following exercises, vectors u and v are...Ch. 2.4 - In the following exercises, vectors u and v are...Ch. 2.4 - In the following exercises, vectors u and v are...Ch. 2.4 - Calculate the determinant |ijk117203| .Ch. 2.4 - Calculate the determinant |ijk034161| .Ch. 2.4 - For the following exercises, the vector u and v...Ch. 2.4 - For the following exercises, the vector u and v...Ch. 2.4 - Find vector (a2b)c , where...Ch. 2.4 - Find vector c(a+3b) , where...Ch. 2.4 - [T] Use the cross product uv to find the acute...Ch. 2.4 - [T] Use the cross product uv to find the obtuse...Ch. 2.4 - Use the sine and cosine of the angle between two...Ch. 2.4 - Verify Lagrange’s identity uv2=u2v2(uv)2 for...Ch. 2.4 - Nonzero vectors u and v are called collinear if...Ch. 2.4 - Nonzero vectors u and v are called collinear if...Ch. 2.4 - Find the area of the parallelogram with adjacent...Ch. 2.4 - Find the area of the parallelogram with adjacent...Ch. 2.4 - Consider paints A(3,1,2),B(2,1,5), and C(1,2,2)....Ch. 2.4 - Consider points A(2,3,4),B(0,1,2), and C(1,2,0)....Ch. 2.4 - In the following exercises, vectors u,v, and w are...Ch. 2.4 - In the following exercises, vectors u,v, and w are...Ch. 2.4 - In the following exercises, vectors u,v, and w are...Ch. 2.4 - In the following exercises, vectors u,v, and w are...Ch. 2.4 - Find vectors a,b, and c with a triple scalar...Ch. 2.4 - The triple scalar product of vector a,b, and c is...Ch. 2.4 - Consider the parallelepiped with edges OA,OB, and...Ch. 2.4 - Consider points A(,0,0),B(0,,0), and C(0,0,), with...Ch. 2.4 - Let u,v, and w be three-dimensional vectors and c...Ch. 2.4 - Show that vectors u=1,0,8,v=0,1,6, and w=1,9,3...Ch. 2.4 - Nonzero vectors u,v, and w are said to be linearly...Ch. 2.4 - Consider vectors u=1,4,7,v=2,1,4,w=0,9,18, and...Ch. 2.4 - Consider points A(0,0,2),B(1,0,2),C(1,1,2), and...Ch. 2.4 - Show that vectors i+j,ij, and i+j+k are linearly...Ch. 2.4 - Let u=u1,u2,0 and v=v1,v2,0, be two-dimensional...Ch. 2.4 - Consider points P(2,1),Q(4,2) and R(1,2). Find the...Ch. 2.4 - Determine a vector of magnitude 10 perpendicular...Ch. 2.4 - Determine a unit vector perpendicular to the plane...Ch. 2.4 - Consider u and v two three-dimensional vectors. If...Ch. 2.4 - [T] Assume that the magnitudes of two nonzero...Ch. 2.4 - Find all vectors w=w1,w2,w3 that satisfy the...Ch. 2.4 - - Solve the w1,0,1=3,0,3, where w=w1,w2,w3 is a...Ch. 2.4 - [T] A mechanic uses a 12in . wrench to turn a...Ch. 2.4 - [T] A boy applies the brakes on a bicycle by...Ch. 2.4 - [T] Find the magnitude of the force that needs to...Ch. 2.4 - [T] What is the magnitude of the force required to...Ch. 2.4 - [T] The fence vector F acting on a proton with all...Ch. 2.4 - [T] The fence vector F acting on a proton with an...Ch. 2.4 - [T] Consider r(t)=cost,sint,2t the position vector...Ch. 2.4 - A solar panel is mounted 011 the roof of a house....Ch. 2.5 - Figure 2.73 Industrial pipe installations often...Ch. 2.5 - Figure 2.73 Industrial pipe installations often...Ch. 2.5 - Figure 2.73 Industrial pipe installations often...Ch. 2.5 - Figure 2.73 Industrial pipe installations often...Ch. 2.5 - Figure 2.73 Industrial pipe installations often...Ch. 2.5 - Figure 2.73 Industrial pipe installations often...Ch. 2.5 - Figure 2.73 Industrial pipe installations often...Ch. 2.5 - Figure 2.73 Industrial pipe installations often...Ch. 2.5 - Figure 2.73 Industrial pipe installations often...Ch. 2.5 - In the following exercises, points P and Q are...Ch. 2.5 - In the following exercises, points P and Q are...Ch. 2.5 - In the following exercises, points P and Q are...Ch. 2.5 - In the following exercises, points P and Q are...Ch. 2.5 - For the following exercises, point P and vector v...Ch. 2.5 - For the following exercises, point P and vector v...Ch. 2.5 - For the following exercises, point P and vector v...Ch. 2.5 - For the following exercises, point P and vector v...Ch. 2.5 - For the following exercises, line L is given. Find...Ch. 2.5 - For the following exercises, line L is given. Find...Ch. 2.5 - For the following exercises, line L is given. Find...Ch. 2.5 - For the following exercises, line L is given. Find...Ch. 2.5 - For the following exercises, lines L1 and L2 are...Ch. 2.5 - For the following exercises, lines L1 and L2 are...Ch. 2.5 - For the following exercises, lines L1 and L2 are...Ch. 2.5 - For the following exercises, lines L1 and L2 are...Ch. 2.5 - For the following exercises, lines L1 and L2 are...Ch. 2.5 - For the following exercises, lines L1 and L2 are...Ch. 2.5 - For the following exercises, Lines L1 and L2 are...Ch. 2.5 - For the following exercises, Lines L1 and L2 are...Ch. 2.5 - For the following exercises, Lines L1 and L2 are...Ch. 2.5 - For the following exercises, Lines L1 and L2 are...Ch. 2.5 - For the following exercises, Lines L1 and L2 are...Ch. 2.5 - For the following exercises, Lines L1 and L2 are...Ch. 2.5 - For the following exercises, point P and vector n...Ch. 2.5 - For the following exercises, point P and vector n...Ch. 2.5 - For the following exercises, point P and vector n...Ch. 2.5 - For the following exercises, point P and vector n...Ch. 2.5 - For the following exercises, the equation of a...Ch. 2.5 - For the following exercises, the equation of a...Ch. 2.5 - For the following exercises, the equation of a...Ch. 2.5 - For the following exercises, the equation of a...Ch. 2.5 - Given paint P(1,2,3) and vector n=i+j , find point...Ch. 2.5 - Show there is no plane perpendicular to n=i+j that...Ch. 2.5 - Find parametric equations 0f the Line passing...Ch. 2.5 - Find symmetric equations of the line passing...Ch. 2.5 - Show that line x12=y+13=z24 is parallel to plane...Ch. 2.5 - Find the real number such that the line of...Ch. 2.5 - For the following exercises, points P,Q, and R are...Ch. 2.5 - For the following exercises, points P,Q, and R are...Ch. 2.5 - Consider the planes of equations x+y+z=1 and x+z=0...Ch. 2.5 - Consider the planes of equations y+z2=0 and xy=0 ....Ch. 2.5 - Find the scalar equation of the plane that passes...Ch. 2.5 - Find the general equation of the plane that passes...Ch. 2.5 - Determine whether the Line of parametric equations...Ch. 2.5 - Determine whether the Line of parametric equations...Ch. 2.5 - Find the distance from point P(1,5,4) to the plane...Ch. 2.5 - Find the distance from point P(1,2,3) to the plane...Ch. 2.5 - For the following exercises, the equations of two...Ch. 2.5 - For the following exercises, the equations of two...Ch. 2.5 - For the following exercises, the equations of two...Ch. 2.5 - For the following exercises, the equations of two...Ch. 2.5 - Show that the lines of equations x=t,y=1+t,z=2+t,t...Ch. 2.5 - Show that the lines of equations...Ch. 2.5 - Consider point C(3,2,4) and the plane of equation...Ch. 2.5 - Consider the plane of equation xyz8=0. Find the...Ch. 2.5 - Two children are playing with a ball. The girl...Ch. 2.5 - [T] John allocates d dollars to consume monthly...Ch. 2.5 - [T] Consider r(t)=sint,cost,2t the position vector...Ch. 2.5 - [T] A solar panel is mounted on the roof of a...Ch. 2.6 - For the following exercises, sketch and describe...Ch. 2.6 - For the following exercises, sketch and describe...Ch. 2.6 - For the following exercises, sketch and describe...Ch. 2.6 - For the following exercises, sketch and describe...Ch. 2.6 - For the following exercises, sketch and describe...Ch. 2.6 - For the following exercises, sketch and describe...Ch. 2.6 - For the following exercises, the graph of a...Ch. 2.6 - For the following exercises, the graph of a...Ch. 2.6 - For the following exercises, the graph of a...Ch. 2.6 - For the following exercises, the graph of a...Ch. 2.6 - For the fallowing exercises, match the given...Ch. 2.6 - For the fallowing exercises, match the given...Ch. 2.6 - For the fallowing exercises, match the given...Ch. 2.6 - For the fallowing exercises, match the given...Ch. 2.6 - For the fallowing exercises, match the given...Ch. 2.6 - For the fallowing exercises, match the given...Ch. 2.6 - For the following exercises, rewrite the given...Ch. 2.6 - For the following exercises, rewrite the given...Ch. 2.6 - For the following exercises, rewrite the given...Ch. 2.6 - For the following exercises, rewrite the given...Ch. 2.6 - For the following exercises, rewrite the given...Ch. 2.6 - For the following exercises, rewrite the given...Ch. 2.6 - For the following exercises, rewrite the given...Ch. 2.6 - For the following exercises, rewrite the given...Ch. 2.6 - For the following exercises, rewrite the given...Ch. 2.6 - For the following exercises, rewrite the given...Ch. 2.6 - For the following exercises, rewrite the given...Ch. 2.6 - For the following exercises, rewrite the given...Ch. 2.6 - For the following exercises, find the trace of the...Ch. 2.6 - For the following exercises, find the trace of the...Ch. 2.6 - For the following exercises, find the trace of the...Ch. 2.6 - For the following exercises, find the trace of the...Ch. 2.6 - For the following exercises, find the trace of the...Ch. 2.6 - For the following exercises, find the trace of the...Ch. 2.6 - Use the graph of the given quadric surface to...Ch. 2.6 - Use the graph of the given quadric surface to...Ch. 2.6 - For the following exercises, the equation of a...Ch. 2.6 - For the following exercises, the equation of a...Ch. 2.6 - For the following exercises, the equation of a...Ch. 2.6 - For the following exercises, the equation of a...Ch. 2.6 - For the following exercises, the equation of a...Ch. 2.6 - For the following exercises, the equation of a...Ch. 2.6 - Write the standard form of the equation of the...Ch. 2.6 - Write the standard form of the equation of the...Ch. 2.6 - Determine the intersection points of elliptic cone...Ch. 2.6 - Determine the intersection points of parabolic...Ch. 2.6 - Find the equation of the quadric surface with...Ch. 2.6 - Find the equation of the quadric surface with...Ch. 2.6 - If the surface of a parabolic reflector is...Ch. 2.6 - Consider the parabolic reflector described by...Ch. 2.6 - Show that quadric surface...Ch. 2.6 - Show that quadric surface x2+y2+z22xy+2yz1=0...Ch. 2.6 - [T] The intersection between cylinder (x1)2+y2=1...Ch. 2.6 - Hyperboloid of one sheet 25x2+25y2z2=25 and...Ch. 2.6 - [T] Use a CA8 to create the intersection between...Ch. 2.6 - [T] A spheroid is an ellipsoid with two equal...Ch. 2.6 - [T] In cartography, Earth is approximated by all...Ch. 2.6 - [T] A set 0f buzzing stunt magnets (or...Ch. 2.6 - [T] A heart-shaped surface is given by equation...Ch. 2.6 - [T] The ring torus symmetric about the z -axis is...Ch. 2.7 - Use the following ?gure as an aid in identifying...Ch. 2.7 - Use the following ?gure as an aid in identifying...Ch. 2.7 - Use the following ?gure as an aid in identifying...Ch. 2.7 - Use the following ?gure as an aid in identifying...Ch. 2.7 - For the following exercises, the rectangular...Ch. 2.7 - For the following exercises, the rectangular...Ch. 2.7 - For the following exercises, the rectangular...Ch. 2.7 - For the following exercises, the rectangular...Ch. 2.7 - For the following exercises, the equation of a...Ch. 2.7 - For the following exercises, the equation of a...Ch. 2.7 - For the following exercises, the equation of a...Ch. 2.7 - For the following exercises, the equation of a...Ch. 2.7 - For the following exercises, the equation of a...Ch. 2.7 - For the following exercises, the equation of a...Ch. 2.7 - For the following exercises, the equation of a...Ch. 2.7 - For the following exercises, the equation of a...Ch. 2.7 - For the following exercises, the equation of a...Ch. 2.7 - For the following exercises, the equation of a...Ch. 2.7 - For the following exercises, the equation of a...Ch. 2.7 - For the following exercises, the equation of a...Ch. 2.7 - For the following exercises, the equation of a...Ch. 2.7 - For the following exercises, the equation of a...Ch. 2.7 - For the following exercises, the spherical...Ch. 2.7 - For the following exercises, the spherical...Ch. 2.7 - For the following exercises, the spherical...Ch. 2.7 - For the following exercises, the spherical...Ch. 2.7 - For the following exercises, the rectangular...Ch. 2.7 - For the following exercises, the rectangular...Ch. 2.7 - For the following exercises, the rectangular...Ch. 2.7 - For the following exercises, the rectangular...Ch. 2.7 - For the following exercises, the equation of a...Ch. 2.7 - For the following exercises, the equation of a...Ch. 2.7 - For the following exercises, the equation of a...Ch. 2.7 - For the following exercises, the equation of a...Ch. 2.7 - For the following exercises, the equation of a...Ch. 2.7 - For the following exercises, the equation of a...Ch. 2.7 - For the following exercises, the equation of a...Ch. 2.7 - For the following exercises, the equation of a...Ch. 2.7 - For the following exercises, the equation of a...Ch. 2.7 - For the following exercises, the equation of a...Ch. 2.7 - For the following exercises, the cylindrical...Ch. 2.7 - For the following exercises, the cylindrical...Ch. 2.7 - 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- R denotes the field of real numbers, Q denotes the field of rationals, and Fp denotes the field of p elements given by integers modulo p. You may refer to general results from lectures. Question 1 For each non-negative integer m, let R[x]m denote the vector space consisting of the polynomials in x with coefficients in R and of degree ≤ m. x²+2, V3 = 5. Prove that (V1, V2, V3) is a linearly independent (a) Let vi = x, V2 = list in R[x] 3. (b) Let V1, V2, V3 be as defined in (a). Find a vector v € R[×]3 such that (V1, V2, V3, V4) is a basis of R[x] 3. [8] [6] (c) Prove that the map ƒ from R[x] 2 to R[x]3 given by f(p(x)) = xp(x) — xp(0) is a linear map. [6] (d) Write down the matrix for the map ƒ defined in (c) with respect to the basis (2,2x + 1, x²) of R[x] 2 and the basis (1, x, x², x³) of R[x] 3. [5]arrow_forwardQuestion 4 (a) The following matrices represent linear maps on R² with respect to an orthonormal basis: = [1/√5 2/√5 [2/√5 -1/√5] " [1/√5 2/√5] A = B = [2/√5 1/√5] 1 C = D = = = [ 1/3/5 2/35] 1/√5 2/√5 -2/√5 1/√5' For each of the matrices A, B, C, D, state whether it represents a self-adjoint linear map, an orthogonal linear map, both, or neither. (b) For the quadratic form q(x, y, z) = y² + 2xy +2yz over R, write down a linear change of variables to u, v, w such that q in these terms is in canonical form for Sylvester's Law of Inertia. [6] [4]arrow_forwardpart b pleasearrow_forward
- Question 5 (a) Let a, b, c, d, e, ƒ Є K where K is a field. Suppose that the determinant of the matrix a cl |df equals 3 and the determinant of determinant of the matrix a+3b cl d+3e f ГЪ e [ c ] equals 2. Compute the [5] (b) Calculate the adjugate Adj (A) of the 2 × 2 matrix [1 2 A = over R. (c) Working over the field F3 with 3 elements, use row and column operations to put the matrix [6] 0123] A = 3210 into canonical form for equivalence and write down the canonical form. What is the rank of A as a matrix over F3? 4arrow_forwardQuestion 2 In this question, V = Q4 and - U = {(x, y, z, w) EV | x+y2w+ z = 0}, W = {(x, y, z, w) € V | x − 2y + w − z = 0}, Z = {(x, y, z, w) € V | xyzw = 0}. (a) Determine which of U, W, Z are subspaces of V. Justify your answers. (b) Show that UW is a subspace of V and determine its dimension. (c) Is VU+W? Is V = UW? Justify your answers. [10] [7] '00'arrow_forwardGood explanation it sure experts solve itarrow_forward
- Best explains it not need guidelines okkarrow_forwardTask number: A1.1, A1.7 Topic: Celestial Navigation, Compass - Magnetic and Gyro Activ Determine compass error (magnetic and gyro) using azimuth choosing a suitable celestial body (Sun/ Stars/ Planets/ Moon). Apply variation to find the deviation of the magnetic compass. Minimum number of times that activity should be recorded: 6 (2 each phase) Sample calculation (Azimuth- Planets): On 06th May 2006 at 22h20m 10s UTC, a vessel in position 48°00'N 050°00'E observed Mars bearing 327° by compass. Find the compass error. If variation was 4.0° East, calculate the deviation. GHA Mars (06d 22h): Increment (20m 10s): 089° 55.7' 005° 02.5' v (0.9): (+) 00.3' GHA Mars: 094° 58.5' Longitude (E): (+) 050° 00.0' (plus- since longitude is easterly) LHA Mars: 144° 58.5' Declination (06d 22h): d (0.2): N 024° 18.6' (-) 00.1' Declination Mars: N 024° 18.5' P=144° 58.5' (If LHA<180°, P=LHA) A Tan Latitude/ Tan P A Tan 48° 00' Tan 144° 58.5' A = 1.584646985 N (A is named opposite to latitude, except when…arrow_forwardTask number: A1.1, A1.7 Topic: Celestial Navigation, Compass - Magnetic and Gyro Activ Determine compass error (magnetic and gyro) using azimuth choosing a suitable celestial body (Sun/ Stars/ Planets/ Moon). Apply variation to find the deviation of the magnetic compass. Minimum number of times that activity should be recorded: 6 (2 each phase) Sample calculation (Azimuth- Planets): On 06th May 2006 at 22h20m 10s UTC, a vessel in position 48°00'N 050°00'E observed Mars bearing 327° by compass. Find the compass error. If variation was 4.0° East, calculate the deviation. GHA Mars (06d 22h): Increment (20m 10s): 089° 55.7' 005° 02.5' v (0.9): (+) 00.3' GHA Mars: 094° 58.5' Longitude (E): (+) 050° 00.0' (plus- since longitude is easterly) LHA Mars: 144° 58.5' Declination (06d 22h): d (0.2): N 024° 18.6' (-) 00.1' Declination Mars: N 024° 18.5' P=144° 58.5' (If LHA<180°, P=LHA) A Tan Latitude/ Tan P A Tan 48° 00' Tan 144° 58.5' A = 1.584646985 N (A is named opposite to latitude, except when…arrow_forward
- Activ Determine compass error using amplitude (Sun). Minimum number of times that activity should be performed: 3 (1 each phase) Sample calculation (Amplitude- Sun): On 07th May 2006 at Sunset, a vessel in position 10°00'N 010°00'W observed the Sun bearing 288° by compass. Find the compass error. LMT Sunset: LIT: (+) 00d 07d 18h 00h 13m 40m UTC Sunset: 07d 18h 53m (added- since longitude is westerly) Declination (07d 18h): N 016° 55.5' d (0.7): (+) 00.6' Declination Sun: N 016° 56.1' Sin Amplitude = Sin Declination/Cos Latitude = Sin 016°56.1'/ Cos 10°00' = 0.295780189 Amplitude=W17.2N (The prefix of amplitude is named easterly if body is rising, and westerly if body is setting. The suffix is named same as declination) True Bearing=287.2° Compass Bearing= 288.0° Compass Error = 0.8° Westarrow_forwardOnly sure experts solve it correct complete solutions okkarrow_forward4c Consider the function f(x) = 10x + 4x5 - 4x³- 1. Enter the general antiderivative of f(x)arrow_forward
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