If u and v are vectors in R^n, and if v≠0 ⃗, then u can be expressed in exactly one way in the form u=w_1+w_2, where w_1 is a scalar multiple of v and w_2 is orthogonal to v. The vector w_1 is called the _____.

Q: (1, Find a vector orthogonal to both (3, 2, 0) and to (0, 2, -1) of the form

A: The given question is related with vector space. We have to find a vector orthogonal to both 3 , 2 ,…

Q: Find a vector of norm 1 that is orthogonal to the three vectors ü = (2,1,-4,0), v=(-1,-1,2,2),…

A: We have to find a vector of norm 1 which is orthogonal with the vectors u, v and w. Let a, b, c, d…

Q: Determine a vector that is orthogonal to both (1, 2, -1) and (3, 1, 0).

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Q: Prove that llu - vll >=llull -llvll for all vectors u and v in Rn.

A: According to the given information, it is required to prove the inequality.

Q: Find linearly independent vectors which generate al lutions of Ax = 0.

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Q: If a vector is multiplied by a positive scalar, how is the result re-lated to the original vector?…

A: Explanation:

Q: 6. List 3 non-zero vectors in R° that are orthogonal. (You must verify that they are orthogonal.)

A: See the detailed solution below.

Q: Find all vectors in R' that are perpendicular to both [1,0, 1, 0] and [1,2,0, 1] and then find a…

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Q: Find the components of the vector whose initial point is (2, 2, 4) and terminal point at (0, 2, 0).

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Q: Show that the vectors ⟨1, 2, 3⟩, ⟨1, 1, 0⟩, and ⟨3, 4, 3⟩ lie in the same plane

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Q: What is the scalar projection of á = (1,0, 1) onto b = (0, 3, 4)?

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Q: Prove that u · v = 1/4 llu + vll2 -1/4llu - vll2 for all 4 4 vectors u and v in Rn.

A: Given Data The equation to prove is u∙v=1/4||u+v||²-1/4||u-v||².   Evaluate the right hand side for…

Q: Find the vector projection of (3, 3, 4) onto (–1, -2, 2).

A: Projection of u onto v is given by.

Q: 3. Give two nonparallel vectors that are both orthogonal to (0, 5, 0).

A: Given A = (0,5,0) Let other two vectors be of type B = (a,b,c) A.B = a.0+b.5+c.0 A.B = 5b For…

Q: Find a vector of length 5 that is orthogonal to both vectors ⟨4,−1,0⟩ and ⟨−3,2,−1⟩

A: The vector orthogonal to two vectors u,v is = ± (u×v)/ |u| |v| Here u =  ⟨4,−1,0⟩ , v =…

Q: make an example and prove it about five vectors spanning R⁴

A: Given: Five vector spanning R4  We need to form an example and prove the given statement 'Five…

Q: A type (0,4) tensor Tabed satisfies Tabcd A® µ°A°µ¢ = 0 for all con- travariant vectors X and uº.…

A: Tensor of type (0, 4): A tensor of the form Tabcd is called covariant tensor of type (0, 4). A…

Q: The vectors in Z22 are [ O, O] , [ O, l], [l, O ], and [l, l ]. (How many vectors doesZn 2 contain,…

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Q: Find the magnitude of the projection of (8, – 2) onto the vector (4, 7)

A: Wr have to find the magnitude of the projection.

Q: Let a = (7, 10, 1) and b = (10, 3, 6) be vectors. Compute the cross product %3D ахЬ.

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Q: 2) How many different vectors are drawn here? -/

A: 3 vectors are same which has same direction and magnitude and other 2 vectors are same in direction…

Q: A vector orthogonal to the plane through the poii (-1,0,0), (0, –2, –2), (2, 3, 4) is < -2, – 10, 9…

A: The given 3 points are A (-1,0,0) , B (0,-2,-2) &amp; C (2,3,4) Let, x1=-1, y1=0, z1=0 x2=0, y2=-2 ,…

Q: Find the component form and length of the vector v that has initial point (3, −7) and terminal point…

A: Given: Initial point (3,-7) and terminal point (-2,5).

Q: Show that the vectors are orthogonal.

A: Given vectors are u=&lt;u1, u2&gt;=&lt;4, -1&gt;v=&lt;v1, v2&gt;=&lt;2, 8&gt;

Q: Find a vector orthogonal to both (-3, 2,0) and to (0, 2, 1) of the form (1,

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Q: The vectors [1, 0, 1], [1, 1, 1] and [1, 0, -1] are coplanar. True False

A: Using coplanarity conditions we have

Q: Find two unit vectors orthogonal to both (5, 8, 1) and (-1, 1, 0). (smaller i-value) (larger…

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Q: 1 Defined in P,(R), p(x)=1,q(x)=x- and h(x) =x² -: -x+- 6. Investigate whether the vectors are…

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Q: If the vectors a and b are such that a = kb where k is a positive scalar, then the vectors a and b…

A: Given statement: If the vectors a and b are such that a=kb where k is apositive scalar, then the…

Q: Show that the vectors , and are coplanar.

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Q: b) Express the vector (–1,5, -6) in R³ as a linear combination of (2,0, 7), (2,4, 5) and(2, –12,13).

A: See the detailed solution below.

Q: Show that there are infinitely many vectors in R with Euclidean norm 1 whose Euclidean inner product…

A: Dear students you have asked multiple questions. As per our policy your first question will be…

Q: 8. Show that the cross product u x v of the vectors u = (-3, 5, -4) and v = (-3, -5, -5) is…

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Q: uppose ā and b are two non-zero vectors in R3 and they are not parallel to each other. Then, their…

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Q: A vector orthogonal to the plane through the points (-1,0,0), (0,0, –2), (2,3, 4) is О О O 6,-10, 3>…

A: Let, Given vectors are P(-1,0,0), Q(0,0,-2) and R(2,3,4). We have to find non zero vector orthogonal…

Q: Define whether these vectors are linearly independent! --- u =| 2 3, 12.

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Q: Find the sum of the vectors (3,0,1) and (0,8, 0) and illustrate geometrically.

A: Solved below

Q: (i) Compute each Ax by dot products of the rows with the column vector: (a) 2 1 1 2 3 [10 2 1 1 2 3…

A: Since, you have asked two questions, we are attempting only the first question for now. If you want…

Q: A vector orthogonal to the plane through the points (-1,0,0), (0, 0, -2), (2, 3, 4) is

A: We will solve the problem

Q: Find a vector orthogonal to both (-4,-4,0) and to (0,-4,-4) of the form (100)

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Q: 2. If a set of vectors contains at least 2 vectors that are scalar multiples of each other, then the…

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Q: Show that the vectors (1,1,0), (1,0,1), and (0,1,1) generate F3.

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Q: Find a vector orthogonal to both (-4,5, 0) and to (0, 5, 4) of the form (1,

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Q: 6. Find a unit vector that is orthogonal to both [0, 1, 1] and [1,0, 1].

A: We will find the unit vector orthogonal to both the vectors as following.

Q: Find a vector b such that compab = 2. If a = ⟨3,0,−1⟩

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Q: Show that the vectors are orthogonal.

A: Given : 3,6 and 4,-2

Q: If u and a are vectors in 2-space or 3-space and if a ± 0, then proju= u - a 12 (vector component of…

A: Let a and u be any vector in space and θ is the angle between them as shown below: From the above…

Q: Find the unit vectoru in the di rection of the vectorv=3(2, 2, 1).

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