5. Let V = R³3 and let (, ·) be the standard dot prod-uct.(a) Calculate (3e, + 2e, – e3, 2e2 + 4e3).(b) Let W = Span(e, + e, + e3) and find W+.(c) Using the Gram Schmidt process, find an or-thogonal basis for V containing the vectore1 – E2 + 2e3.

Assuming the given vectors e1, e2 and e3 are standard basis vectors. <e_i , e_j>=0 for i≠j…

Answered: 5. Let V = R3 and let (, ·) be the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Find the decomposition of a(t) into tangential and normal components at the point indicated. = (3-1₁t+5,1²), t = 2 (Give your answer using component form or standard basis vectors. Express numbers in exact form. Use symbolic notation and fractions where needed.) ATT = Incorrect aNN Incorrect = 0,0, 16t √18
6. Project the vector b = (1, 4, 2) onto the plane S that contains the vectors a₁ = (1, 2, 1) and a2 = (1, 1,0). (i) Find the 3 x 3 projection matrix P onto S. (ii) Show that P2 = P. (iii) Find a vector whose projection onto S is the zero vector. (iv) Show that I - P projects onto the line perpendicular to S.
The coordinate vector of the vector (-4,3,1) in the basis B={u= (1,1,1); v = (1,0,1); w= (0,0,1) } is: OA (2,-1,1) OB. (3,-7,5) OC. (1,-4,3) OD. (-4,3,1)
Let a = (-6, 2, 7) and b = (10, –1,6) be vectors. (A) Find the scalar projection of b onto a. Scalar Projection: (B) Decompose the vector b into a component parallel to a and a component orthogonal to a. Parallel component: ( Orthogonal Component: (
2. The vector x = H [13] vector [2]B = [8] A 1 B 2 3 D0 in R2 relative to the orthogonal basis B What is the value of a + b? {0-[-]} has coordinate
Find the coordinate vector of v relative to the basis S = {v1, V2, V3 } for R3. v = (41, 39, 21); v1 = (7, 7, 7). (2, 0, 0), v2 = (6, 6, 0), v3 = (v)s=( i i i ).
Compute the curl of each of the following vectors: V = (yz, xz, xy) F = (x, y, z) = = ( Note: Y 2 (x² + y² +2²)³/²¹ (x² + y² + x2)³/²¹ (x² + y² + x2)³/2 • If the answer is a scalar, you can just type it in the box (using the Calcpad if you like, or using / for fractions, ^ for exponents, etc.) For multiplication, you can either leave a space, or use *. So x*x=xx = x². Note that this is not the same as xx without a space; that will get read as an entirely different variable! • If you need to enter a vector, enter an ordered list of components, so for A you can enter either A = (A₂, Ay, A₂) = {Ax, Ay, Az} . Note that the system isn't great with multiplying through by overall factors, so it's better not to write e.g. • If you need to enter O, enter it in three components, as (0, 0, 0) or {0,0,0}. V x V = ▼ xr= vx= (2A, 2A, 2A₂).
Find an orthonormal basis of the plane r + 7r2 - r3 = 0.

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5. Let V = R3 and let (, ·) be the standard dot prod- uct. (a) Calculate (3e, + 2e, – e3, 2e, + 4e3). (b) Let W Span(e, + e, + e3) and find W+. (c) Using the Gram Schmidt process, find an or- thogonal basis for V containing the vector ej – Eg + 2e3. |