For any geometric vectors u and v, prove the following using triangle inequality. (a) ||u – v|| < ||u| + ||v|| (b) u|| - ||v|| < ||u – ||

Q: Two vectors u and v in space are if and only if u X v= 0

A: Given vectors in space are u and v The vectors cross product is u x v = |u| * |v|* sin(angle between…

Q: Among all unit vectors u = in R°, find the one for which the sum x + 6y + 9z is minimal. నా

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Q: Q2 (a) Let n E N and let 7, w e R". Show that if ở - = 0, then || † + w||? = || T||? + ||||?. (b)…

A: To answer the given question we shall use the following rule: 1.   If x→∈ℝn, then…

Q: Find the value of k such that vectors x

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Q: Prove the property if a and b are vectors and m is a real number.0 . a = 0

A: Given: If a and b are vectors and m is a real number. To prove: 0·a = 0

Q: 4. Suppose the vectors u and w are in space R, then u - 2w and u+ 2w are perpendicular if (a) ju =…

A: Our guidelines we are supposed to answer only one question. Kindly repost other question as the…

Q: Among all unit vectors u = in R°, find the one for which the sum x + 3y + 8z is minimal.

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Q: Find the length and direction (when defined) of u x v. u = 2 i+ 2j- k, v = -i+ k

A: As per our guidelines we are supposed to answer only one question and rest of the questions are…

Q: Which of the following can NOT be proven using the Cauchy Inequality. O If x and y are vectors in…

A: Answer

Q: -Let 1+t and tet? be vectors in P. write the vector plt) = S+ St+2t² as the lineor combination of P,…

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Q: The cross product is not associative. (It is not necessary for | a xb |x c X c to be the x c |.)…

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Q: Three vectors u, v, and w exist such that - + = 0. Determine the value of ū+uw + (-). w, given that…

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Q: Г 26 Let W be the set of all vectors of the form Find vectors i and v in R such that W = span {u,…

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Q: \(\) The vectors u= and V = are not orthogonal if and only if x=-136

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Q: Let v and w be vectors in R° such that v · w = Which of the following is a possible angle between v…

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Q: Q2 (a) Let n E N and let ở, w e R". Show that if ở · w = 0, then || ở + w||2 = || ||² + ||w||². (b)…

A: This question can be solved using the concept of dot product of two vectors.

Q: (b) For any arbitrary vectors u, v and w, prove that (u x v) x w = u x (v x w) if and only if (u x…

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Q: For each statement, determine whether it is true or false, and justify your af a) In R³ if two lines…

A: (a) False  (b) True  (c) False

Q: Find the value of k for which the vectors 4 and 4 k are orthogonal. k =

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Q: \(\) The vectors u= and U= are not orthogonal if and only if x=-136 Select one: O True O False

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Q: Given the following vectors →u and →v, find a vector →w in ℝ3 so that {→u, →v, →w} spans ℝ3 and a…

A: Now u = {2 ,4,10} and v = {5 , -6 , 7}  We can see that u and v vectors are linearly independent.…

Q: d) Prove that ||vxw||²+|v-w[? = ||v||²|w||² for all vectors v, w in R3

A: Prove the following

Q: 2. The angle e between the vectors u and v satisfies cos 0 = So if u and v are perpendicular, then…

A: The angle between two vectors is, cosθ=u·vu v The vectors are perpendicular if u·v=0. The given…

Q: (b) 4x Find any values of x such that the vectors 3 and are parallel X-1 to each other.

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Q: The vectors u and v are ________ if and only if u ∙ v = 0.

A: If the dot product of two vectors is zero ,                    If and only if they are orthogonal;…

Q: 1 Find the length of the vector x = -4 -5 Enter 'sqrt(a)' for va. ||F| =

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Q: Two vectors u and v in space are if and only if u .v=0

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Q: Find the maximum and minimum values of the ratio x’A x/x'x for any nonzero vectors x' = [x1, x2, X3)…

A: We have to find maximum and minimum value of ratio x'Ax/x'x

Q: Suppose ~u and ~w are vectors in R^n . Prove that if ~u and ~w are parallel vectors in the same…

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Q: Let u¯¯¯=⟨−3,−5⟩, v¯¯¯=⟨2,−2⟩, and w¯¯¯¯=⟨2,4⟩. Find the vector x¯¯¯ that satisfies 9u−v+x=2x+w

A: Given that: u=-3, -5v=2, -2w=2, 4

Q: H is the set of all vectors x = X2 in R³ with the property that X3 X1 + x2 = 0 and Xi – x2 + x3 = 0.…

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Q: Under what condition for the real numbers a,b the vectors (b,-1)(2,a) are linearly related

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Q: a) Find all the real numbers à such the vectors 1- x 1+x and x+1 1 are orthogonal.

A: Two vectors are said to be orthogonal if their dot product is zero. Dot Product There are two…

Q: Suppose Q : R? → R is a quadratic form such that Q(x) = 1 for all vectors X E R? Cuch that

A: Given: It is given that the quadratic function Q:R2→R such that Q(x)=1 for all vectors x∈R2 such…

Q: Let vi = 2 v2 , V3 = -3 3 Does the set of vectors {v, ,v2, v3} span R3 ? Give reason(s) for your…

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Q: 18. Show by vector methods that the median (a median joins the vertex of a triangle to the midpoint…

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Q: 5b- За Let W be the set of all vectors of the form Find vectors i and v in R such that W = span {i,…

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Q: Let W be the set of all vectors of the form t - s Find vectors u and v in R' such that W = span {u,…

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Q: Let FE R?. Show that F = õ if and only if F. Ã=0 for all vectors Ā E R?. [This is a proof, be sure…

A: Answer

Q: b/Let u and v be a nonzero vectors in 3- space, and let e be the angle between these vectors when…

A: Let u and v be tthe vectors in 3-spaceThen by Lagrange's identity,we haveu×v2 =u2.v2-(u.v)2…

Q: 7. The Triangle Inequality for vectors is |a + b| < |a| + [b|. Write a sentence explaining the…

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Q: let A= and Describe the sed of all vectors b for which the equatiön AX=b is Consitent.

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Q: Two vectors u and v in space are if and only if u .v = 0

A: Condition of dot product.

Q: Find the Euclidean norm of the given vector. v = Hint: 3? = 9, 4? = 16 ||v|| = Ex: 1

A: Given vector

Q: Let x1, x2, and x3 be vectors in R3. If x1 ⊥ x2 and x2 ⊥ x3, is it necessarily true that x1 ⊥ x3?…

A: Orthogonal vector ( Perpendicular vector): two vectors u and v are orthogonal if u.v=0

Q: If Q: R" → R is a quadratic form, explain how to determine the maximum and minimum values that Q…

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Q: For the non-zero vectors A and B to be perpendicular to each other, the necessary and sufficient…

A: Given, For the non-zero vectors A and B to be perpendicular to each other, the necessary and…

Q: 1. Consider the vectors x¡ = (8, 6)' and X2 = (4, –1)' in R?. %3D (a) Determine the length of each…

A: (a)

Q: One of the following vectors lies in span{ 2+ x + x2, 3+x- 2x2} (A) 2+ 2x + 7x? (B) 2 + 2x + 5x? (C)…

A: The linear span of the given vectors is of the form a(2+x+x2)+b(3+x-2x2)

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.