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mathematics
precalculus

Questions and Answers of Precalculus

Find the radius and center of each sphere. 2x2 + 2y2 + 2z2 -8x + 4z = -1
Find the radius and center of each sphere. x2 + y2 + z2 -4x = 0
Find the radius and center of each sphere. x2 + y2 + z2 – 4x + 4y + 2z = 0
Find the radius and center of each sphere. x2 + y2 + z2 + 2x – 2z = -1
Find the radius and center of each sphere. x2 + y2 + z2 + 2x – 2y = 2
Find the equation of a sphere with radius r and center P0.r = 2; P0 = (1,2,2)
Find the equation of a sphere with radius r and center P0.r = 1; P0 = (3,1,1)
In space, the collection of all points that are the same distance from some fixed point is called a sphere. See the illustration. The constant distance is called the radius, and the fixed point is
Consider the double-jointed robotic arm shown in the figure. Let the lower arm be modeled by a = (2,3,4) the middle arm be modeled by b = (1, -1, 3) and the upper arm by c = (4,-1,-2) where
Find the direction angles of each vector. Write each vector in the form of equation (7).v = 2i + 3j - 4k M[(cos a)i + (cos B)j + (cos y)k] (7)
Find the direction angles of each vector. Write each vector in the form of equation (7).v = 3i - 5j + 2k M[(cos a)i + (cos B)j + (cos y)k] (7)
Find the direction angles of each vector. Write each vector in the form of equation (7).v = j + k = ||M[(cos a)i + (cos B)j + (cos y)k] (7) V
Find the direction angles of each vector. Write each vector in the form of equation (7).v = i + j M[(cos a)i + (cos B)j + (cos y)k] (7)
Find the direction angles of each vector. Write each vector in the form of equation (7).v = i - j - k M[(cos a)i + (cos B)j + (cos y)k] (7)
Find the direction angles of each vector. Write each vector in the form of equation (7).v = i + j + k M[(cos a)i + (cos B)j + (cos y)k] (7)
Find the direction angles of each vector. Write each vector in the form of equation (7).v = -6i + 12j + 4k
Find the direction angles of each vector. Write each vector in the form of equation (7).v = 3i - 6j - 2k
Find the dot product v.w and the angle between v and w. v = 3i - 4j + k, w = 6i - 8j + 2k
Find the dot product v.w and the angle between v and w. v = 3i + 4j + k, w = 6i + 8j + 2k
Find the dot product v.w and the angle between v and w. v = i + 3j + 2k, w = i - j + k
Find the dot product v.w and the angle between v and w. v = 3i - j + 2k, w = i + j - k
Find the dot product v.w and the angle between v and w. v = 2i + 2j - k, w = i + 2j + 3k
Find the dot product v.w and the angle between v and w. v = 2i + j - 3k, w = i + 2j + 2k
Find the dot product v.w and the angle between v and w. v = i + j, w = -i + j - k
Find the dot product v.w and the angle between v and w. v = i - j, w = i + j + k
Find the unit vector in the same direction as v. v = 2i - j + k
Find the unit vector in the same direction as v. v = i + j + k
Find the unit vector in the same direction as v. v = -6i + 12j + 4k
Find the unit vector in the same direction as v. v = 3i - 6j - 2k
Find the unit vector in the same direction as v. v = -3j
Find the unit vector in the same direction as v. v = 5i
Find each quantity if and v = 3i - 5j + 2k and w = -2i + 3j - 2k. |v | + |w|
Find each quantity if and v = 3i - 5j + 2k and w = -2i + 3j - 2k. |v| - |w|
Find each quantity if and v = 3i - 5j + 2k and w = -2i + 3j - 2k. |v + w|
Find each quantity if and v = 3i - 5j + 2k and w = -2i + 3j - 2k. |v - w|
Find each quantity if and v = 3i - 5j + 2k and w = -2i + 3j - 2k. 3v - 2w
Find each quantity if and v = 3i - 5j + 2k and w = -2i + 3j - 2k. 2v + 3w
Find |v|. v = 6i + 2j - 2k
Find |v|. v = -2i + 3j - 3k
Find |v|. v = -i - j + k
Find |v|. v = i - j + k
Find |v|. v = -6i + 12j + 4k
Find |v|. v = 3i - 6j - 2k
The vector v has initial point P and terminal point Q. Write v in the form ai + bj + ck; that is, find its position vector.P = (-1, 4, -2); Q = (6, 2, 2)
The vector v has initial point P and terminal point Q. Write v in the form ai + bj + ck; that is, find its position vector.P = (-2, -1, 4); Q = (6, -2, 4)
The vector v has initial point P and terminal point Q. Write v in the form ai + bj + ck; that is, find its position vector.P = (-3, 2, 0); Q = (6, 5, -1)
The vector v has initial point P and terminal point Q. Write v in the form ai + bj + ck; that is, find its position vector.P = (3, 2, -1); Q = (5, 6, 0)
The vector v has initial point P and terminal point Q. Write v in the form ai + bj + ck; that is, find its position vector.P = (0, 0, 0); Q = (-3, -5, 4)
The vector v has initial point P and terminal point Q. Write v in the form ai + bj + ck; that is, find its position vector.P = (0, 0, 0); Q = (3, 4, -1)
Opposite vertices of a rectangular box whose edges are parallel to the coordinate axes are given. List the coordinates of the other six vertices of the box. (-2, -3, 0); (-6, 7, 1)
Opposite vertices of a rectangular box whose edges are parallel to the coordinate axes are given. List the coordinates of the other six vertices of the box. (-1, 0, 2); (4, 2, 5)
Opposite vertices of a rectangular box whose edges are parallel to the coordinate axes are given. List the coordinates of the other six vertices of the box. (5, 6, 1); (3, 8, 2)
Opposite vertices of a rectangular box whose edges are parallel to the coordinate axes are given. List the coordinates of the other six vertices of the box. (1, 2, 3); (3, 4, 5)
Opposite vertices of a rectangular box whose edges are parallel to the coordinate axes are given. List the coordinates of the other six vertices of the box. (0, 0, 0); (4, 2, 2)
Opposite vertices of a rectangular box whose edges are parallel to the coordinate axes are given. List the coordinates of the other six vertices of the box. (0, 0, 0); (2, 1, 3)
Find the distance from to P2 to P2.P1 = (2, -3, -3) and P2 = (4, 1, -1)
Find the distance from to P2 to P2.P1 = (4, -2, -2) and P2 = (3, 2, 1)
Find the distance from to P2 to P2.P1 = (-2,2,3) and P2 = (4,0,-3)
Find the distance from to P2 to P2.P1 = (-1,2, -3) and P2 = (0,-2,1)
Find the distance from to P2 to P2.P1 = (0, 0, 0) and P2 = (1, -2, 3)
Find the distance from to P2 to P2.P1 = (0, 0, 0) and P2 = (4, 1, 2)
Describe the set of points defined by the equation(s). x = 3 and z = 1
Describe the set of points defined by the equation(s). .x = 1 and y = 2
Describe the set of points defined by the equation(s). z = -3
Describe the set of points defined by the equation(s). x = -4
Describe the set of points defined by the equation(s). y = 3
Describe the set of points defined by the equation(s). z = 2
Describe the set of points defined by the equation(s). x = 0
Describe the set of points defined by the equation(s). y = 0
True or False. A vector in space may be described by specifying its magnitude and its direction angles.
True or False. In space, the dot product of two vectors is a positive number.
The sum of the squares of the direction cosines of a vector in space add up to _______.
If v = ai + bj + ck is a vector in space, the scalars a, b, c are called the________ of v.
In space, points of the form lie in a plane called the ______.
The distance d from P1 = (x1, y) to P2 = (x2, y2) is d = _______.
Prove the polarization identity,|u + c|2 - |u – v|2 = 4(u.v)
In the definition of work given in this section, what is the work done if F is orthogonal to AB(vector)
Let v and w denote two nonzero vectors. Show that the vectors |w|v + |v|w and |w|v - |v|w are orthogonal.
Let v and w denote two nonzero vectors. Show that the vector v - αw is orthogonal to w if α = v•w/|w|2
(a) If u and v have the same magnitude, show that u + v and u - v are orthogonal. (b) Use this to prove that an angle inscribed in a semicircle is a right angle (see the figure). -V
Show that the projection of v onto i is (v.i). Then show that we can always write a vector v as v = (v.i)i + (v.j)j
Suppose that v and w are unit vectors. If the angle between v and i is and that between w and i is β used the idea of the dot product v. w to prove thatcos(α - β) = cosαcosβ + sinαsinβ
If v is a unit vector and the angle between v and i is α show that v = cosαi + sinαJ.
Prove property (5), 0•v = 0
Prove the distributive property: u • (v + w) = u • v + u • w
Find the acute angle that a constant unit force vector makes with the positive x-axis if the work done by the force in moving a particle from to equals 2.
A bulldozer exerts 1000 pounds of force to prevent a 5000-pound boulder from rolling down a hill. Determine the angle of inclination of the hill.
Billy and Timmy are using a ramp to load furniture into a truck. While rolling a 250-pound piano up the ramp, they discover that the truck is too full of other furniture for the piano to fit. Timmy
A Pontiac Bonneville with a gross weight of 4500 pounds is parked on a street with a 10° grade. Find the magnitude of the force required to keep the Bonneville from rolling down the hill. What
A Toyota Sienna with a gross weight of 5300 pounds is parked on a street with a grade. See the figure. Find the magnitude of the force required to keep the Sienna from rolling down the hill.
Let the vector R represent the amount of rainfall, in inches, whose direction is the inclination of the rain to a rain gauge. Let the vector A represent the area, in square inches, whose direction is
The amount of energy collected by a solar panel depends on the intensity of the sun’s rays and the area of the panel. Let the vector I represent the intensity, in watts per square centimeter,
A wagon is pulled horizontally by exerting a force of 20 pounds on the handle at an angle of 60° with the horizontal. How much work is done in moving the wagon 100 feet?
Find the work done by a force of 3 pounds acting in the direction 60° to the horizontal in moving an object 6 feet (0,0) to (6,0).
Decompose v into two vectors v1 and v2 where v1 is parallel to w and v2 is orthogonal to w.v = i - 3j, w = 4i - j
Decompose v into two vectors v1 and v2 where v1 is parallel to w and v2 is orthogonal to w.v = 3i + j, w = -2i - j
Decompose v into two vectors v1 and v2 where v1 is parallel to w and v2 is orthogonal to w.v = 2i - j, w = i - 2j
Decompose v into two vectors v1 and v2 where v1 is parallel to w and v2 is orthogonal to w.v = i - j, w = -i - 2j
Decompose v into two vectors v1 and v2 where v1 is parallel to w and v2 is orthogonal to w. v = -3i + 2j, w = 2i + j
Decompose v into two vectors v1 and v2 where v1 is parallel to w and v2 is orthogonal to w.v = 2i - 3j, w = i - j

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