Answered: (x, y) (1.0) (2, 2) (3,0) (0, 1) (-2,2)… | bartleby

Elementary Linear Algebra (MindTap Course List)

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.5: Basis And Dimension

Problem 69E: Find a basis for R2 that includes the vector (2,2).

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Question

Q23. A vector field on R2 is defined by F (x, y) = -y i+xj. Describe F by sketching some of the vectors F
(x, y)
(1.0)
(2, 2)
(3,0)
(0, 1)
(-2,2)
(0, 3)
F(x, y)
(0, 1)
(-2,2)
(0.3)
(-1,0)
(-2,-2)
(-3,0)
F(0,3)
(x, y)
(-1.0)
(-2,-2)
(-3,0)
(0, -1)
(2,-2)
(0, -3)
F(2,2)
SI
F1.0)
F(x, y)
(0.-1)
(2,-2)
(0, -3)
(1,0)
(2, 2)
(3.0)

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Consider the vector v=(1,3,0,4). Find u such that a u has the same direction as v and one-half of its length. b u has the direction opposite that of v and twice its length.
Find a basis for R2 that includes the vector (2,2).
Let S={v1,v2,v3} be a set of linearly independent vectors in R3. Find a linear transformation T from R3 into R3 such that the set {T(v1),T(v2),T(v3)} is linearly dependent.
Consider the vectors u=(6,2,4) and v=(1,2,0) from Example 10. Without using Theorem 5.9, show that among all the scalar multiples cv of the vector v, the projection of u onto v is the closest to u that is, show that d(u,projvu) is a minimum.
13. For the vector fields: A = x + 2y + 3z2; ŷ Check both product rule given in below: B = 3y - 2x ŷ. V(A.B) = A x (V x B) + B × (V x A) + (A · V)B + (B · V)A V. (A x B) = B. (V x A) - A. (V x B)
A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by v=(x-y,z+y+7,z2) and the net is decribed by the equation y=1-x2-z2, y20, and oriented in the positive y- direction. (Use symbolic notation and fractions where needed.)
Compute the vector assigned to the points P = (1, 1) and Q = (-3, -3) by the vector field F = (-y, x). F(1, 1) = F(-3, -3) -
Suppose a, b, and c are positive real numbers satisfying a2 = b2 + c2.r(t) = acos(t),bsin(t),csin(t) , 0 ≤ t ≤ 2πThen the vector-valued functiondescribes a tilted circle (a circle in a plane that is not parallel to one of the coordinateplanes). The center of the circle is O(0, 0, 0).(a) Show that |r(t)| is constant and determine the constant. This is the radius of thecircle.(b) Find an equation for the plane that contains the circle by doing the following:(i) Find three points on the circle.(ii) Use the three points to find a normal vector to the plane containing the three points.(iii) Find an equation for the plane. Check: does the plane contain the center of the circle?(iv) Simplify the equation as much as possible. (Since a, b and c are positive real numbers, you can divide by them without dividing by zero.)
6
For the vector-valued function r(t) = t³i + (2t² + 1)j + (3t − 6)k find the following and express your solution in the standard unit vector notation 3t² i + 4tj + 3k 6ti + 4j + 0k 1. r' (t): - 2.r"(t) 3. r' (t) · r" (t) = 18t³ + 16t 4. r' (t) * r"(t) = −12i - 6tj - 12t²k =
Correct solution will be liked.
No. 6 (a, b, c)

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