4. Use the 3-space standard basis vectors to find three nonzero vectors a, b, c satisfying axb =a x c, where a x b0 and b c.(This proves that the cross product operation does not have cancellation across equations.)

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Answered: 4. Use the 3-space standard basis…

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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Construct an orthonormal set of vectors in ℝ3 from the set of linearly independent vectors ? = {(1,1,1), (1,2,3), (2, −1,1)}.
5. Find the orthogonal projection of the vector b = vectors V₁ = -1- and v2 = onto the subpace E spanned by the (Note that v₁ V₂).
Consider the cross product 'x' of vectors in R3. Is 'x' a binary operation on R3? Is 'x' commutative or associative? Is the dot product of vectors in Rn a binary operation on Rn?
Both a) and b)
7. Find the coordinates of the vector v relative to the ordered basis B. a) B = {x² + 2x + 2, 2x + 3,−x² + x + 1} v=-3x² + 6x + 8 14 D b) B = {[1₁¹] [ = (₁ 3²1 -CAC) = B₂ -QAED = 0 v= c) B₁ v= 22
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).
1) Show that the following three vectors are coplanar. u=(2, 3,-1) v= (0, 1, -2) w=(2, 4, -3).
2a)Use triple scalar to verify that the following vectors are not coplanar. ā=[-2,1,0], b=[-3,4,5], c =[2,0,3] b) Write a = [-4,1,7] as a linear combination of a,b,c
Show that a set of vectors, which contains a set of linearly dependent vectors, is linearly dependent. What is the analogous statement about linearly independent vectors?
Suppose that two vectors z and y in R" are orthogonal, meaning their inner product satisfies z ·g = 0. Show that if both vectors are assumed to be non-zero, then {T, g} are linearly independent. Show that this linear independence fails if at least one of the vectors is the zero vector.
8 a) 4 B) 3 NS) 2nd D) 0 TO) one

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4. Use the 3-space standard basis vectors to find three nonzero vectors a, b, c satisfying axb = a x c, where a × b 0 and b c. (This proves that the cross product operation does not have cancellation across equations.)