(a) To define: The three-dimensional coordinate system and the coordinate planes.

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Answer 1

Question

System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.

Chapter 9.CR, Problem 4CC

To determine

(a)

To define:

The three-dimensional coordinate system and the coordinate planes.

To determine

(b)

To find:

The distance from the point (3,−2,5) to each of the coordinate planes

To determine

(c)

To state:

The formula for the distance between the points P(x1,y1,z1) and Q(x2,y2,z2).

To determine

(d)

To find:

The distance between the points P(1,2,3) and Q(3,−1,4).

To determine

(e)

To state:

The equation of a sphere with centre C(h,k,l) and radius r.

To determine

(f)

To find:

The equation for the sphere of radius 5 centered at the point (1,2,−3).

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1. (20 pts) Determine whether the following statements are true (T) or false (F)? (A reasoning is required.) (1) Let V be the set of all ordered pairs of real numbers. Consider the following addition and scalar multiplication operations on u = u= (u1, u2) and v = (v1, v2): u + v = (U₁ + V₁, U₂ + v₂), ku = (ku₁, u₂). Is V a vector space under the above operations? U2 (2) The set Mmxn of all m×n matrices with the usual operations of addition and scalar multiplication is a vector space. α (3) The dimension of the vector space of all matrices A = [a b] in R2×2 with a+d=0 is 4. (4) The coordinate vector of p(x) = 2-x+x² in P3 relative to the basis S = {1, 1+x, x + x2} is [4 -2 1]. (5) If a 6×4 matrix A has a rank 3, then the dimension of N(A) is 3.

5. (20%) The linear transformation L: P3 → P2 defined by L(f(x)) = f'(x)+ f(0). (a) Find the representing matrix A of L with respect to the ordered basis {x2, x, 1} for P3, and the ordered basis {2,1 - x} for P2. (b) Find the coordinates of the f(x) = 2x² +2 in P3 with respect to the ordered basis {x2,-x, 1}, and find the coordinates of L(f(x)) with respect to the ordered basis {2,1-x}

For the spinner below, assume that the pointer can never lie on a borderline. Find the following probabilities. (enter the probabilities as fractions)

Answer 2

Question

System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.

Chapter 9.CR, Problem 4CC

To determine

(a)

To define:

The three-dimensional coordinate system and the coordinate planes.

To determine

(b)

To find:

The distance from the point (3,−2,5) to each of the coordinate planes

To determine

(c)

To state:

The formula for the distance between the points P(x1,y1,z1) and Q(x2,y2,z2).

To determine

(d)

To find:

The distance between the points P(1,2,3) and Q(3,−1,4).

To determine

(e)

To state:

The equation of a sphere with centre C(h,k,l) and radius r.

To determine

(f)

To find:

The equation for the sphere of radius 5 centered at the point (1,2,−3).

Expert Solution & Answer

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Check out a sample textbook solution

See solution

Answer 3

Question

System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.

Chapter 9.CR, Problem 4CC

To determine

(a)

To define:

The three-dimensional coordinate system and the coordinate planes.

To determine

(b)

To find:

The distance from the point (3,−2,5) to each of the coordinate planes

To determine

(c)

To state:

The formula for the distance between the points P(x1,y1,z1) and Q(x2,y2,z2).

To determine

(d)

To find:

The distance between the points P(1,2,3) and Q(3,−1,4).

To determine

(e)

To state:

The equation of a sphere with centre C(h,k,l) and radius r.

To determine

(f)

To find:

The equation for the sphere of radius 5 centered at the point (1,2,−3).

Expert Solution & Answer

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Check out a sample textbook solution

See solution

Answer 4

Question

System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.

Chapter 9.CR, Problem 4CC

To determine

(a)

To define:

The three-dimensional coordinate system and the coordinate planes.

To determine

(b)

To find:

The distance from the point (3,−2,5) to each of the coordinate planes

To determine

(c)

To state:

The formula for the distance between the points P(x1,y1,z1) and Q(x2,y2,z2).

To determine

(d)

To find:

The distance between the points P(1,2,3) and Q(3,−1,4).

To determine

(e)

To state:

The equation of a sphere with centre C(h,k,l) and radius r.

To determine

(f)

To find:

The equation for the sphere of radius 5 centered at the point (1,2,−3).

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 5

Chapter 9.CR, Problem 4CC

To determine

(a)

To define:

The three-dimensional coordinate system and the coordinate planes.

To determine

(b)

To find:

The distance from the point (3,−2,5) to each of the coordinate planes

To determine

(c)

To state:

The formula for the distance between the points P(x1,y1,z1) and Q(x2,y2,z2).

To determine

(d)

To find:

The distance between the points P(1,2,3) and Q(3,−1,4).

To determine

(e)

To state:

The equation of a sphere with centre C(h,k,l) and radius r.

To determine

(f)

To find:

The equation for the sphere of radius 5 centered at the point (1,2,−3).

Answer 6

Chapter 9.CR, Problem 4CC

To determine

(a)

To define:

The three-dimensional coordinate system and the coordinate planes.

Answer 7

Chapter 9.CR, Problem 4CC

To determine

(a)

To define:

The three-dimensional coordinate system and the coordinate planes.

Answer 8

To determine

(b)

To find:

The distance from the point (3,−2,5) to each of the coordinate planes

Answer 9

To determine

(b)

To find:

The distance from the point (3,−2,5) to each of the coordinate planes

Answer 10

To determine

(c)

To state:

The formula for the distance between the points P(x1,y1,z1) and Q(x2,y2,z2).

Answer 11

To determine

(c)

To state:

The formula for the distance between the points P(x1,y1,z1) and Q(x2,y2,z2).

Answer 12

To determine

(d)

To find:

The distance between the points P(1,2,3) and Q(3,−1,4).

Answer 13

To determine

(d)

To find:

The distance between the points P(1,2,3) and Q(3,−1,4).

Answer 14

To determine

(e)

To state:

The equation of a sphere with centre C(h,k,l) and radius r.

Answer 15

To determine

(e)

To state:

The equation of a sphere with centre C(h,k,l) and radius r.

Answer 16

To determine

(f)

To find:

The equation for the sphere of radius 5 centered at the point (1,2,−3).

Answer 17

To determine

(f)

To find:

The equation for the sphere of radius 5 centered at the point (1,2,−3).

Answer 18

Expert Solution & Answer

Answer 19

Expert Solution & Answer

Answer 20

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Answer 21

Ch. 9.1 - Prob. 1E Ch. 9.1 - CONCEPTS 2. a The length of a vector w=a1,a2 is...Ch. 9.1 - 38 Sketching Vectors Sketch the vector indicated....Ch. 9.1 - Prob. 4E Ch. 9.1 - Prob. 5E Ch. 9.1 - Prob. 6E Ch. 9.1 - SKILLS 3-8 Sketching Vectors Sketch the vector...Ch. 9.1 - Prob. 8E Ch. 9.1 - Prob. 9E Ch. 9.1 - Prob. 10E

(a) To define: The three-dimensional coordinate system and the coordinate planes.

Solution Summary: The author explains the three-dimensional coordinate system and the coordinate planes.

Videos

(a)

To define:

The three-dimensional coordinate system and the coordinate planes.

(b)

To find:

The distance from the point (3,−2,5) to each of the coordinate planes

(c)

To state:

The formula for the distance between the points P(x1,y1,z1) and Q(x2,y2,z2).

(d)

To find:

The distance between the points P(1,2,3) and Q(3,−1,4).

(e)

To state:

The equation of a sphere with centre C(h,k,l) and radius r.

(f)

To find:

The equation for the sphere of radius 5 centered at the point (1,2,−3).

Want to see the full answer?

Chapter 9 Solutions

(a) To define: The three-dimensional coordinate system and the coordinate planes.

Solution Summary: The author explains the three-dimensional coordinate system and the coordinate planes.

Videos

(a) To define: The three-dimensional coordinate system and the coordinate planes.

(b) To find: The distance from the point (3,−2,5) to each of the coordinate planes

(c) To state: The formula for the distance between the points P(x1,y1,z1) and Q(x2,y2,z2).

(d) To find: The distance between the points P(1,2,3) and Q(3,−1,4).

(e) To state: The equation of a sphere with centre C(h,k,l) and radius r.

(f) To find: The equation for the sphere of radius 5 centered at the point (1,2,−3).

Want to see the full answer?

Chapter 9 Solutions

(a)

To define:

The three-dimensional coordinate system and the coordinate planes.

(b)

To find:

The distance from the point (3,−2,5) to each of the coordinate planes

(c)

To state:

The formula for the distance between the points P(x1,y1,z1) and Q(x2,y2,z2).

(d)

To find:

The distance between the points P(1,2,3) and Q(3,−1,4).

(e)

To state:

The equation of a sphere with centre C(h,k,l) and radius r.

(f)

To find:

The equation for the sphere of radius 5 centered at the point (1,2,−3).