3 3 17) Let T be a linear transformation from R° to R, andu and v be orthogonal vectors. Which of the following pairs of vectors must be orthogonal? a) Tu and u b) Tv and v Tu and Tv c) d) ru and sv (where r and s are real numbers) e) u+v and u-v

Q: ii) Show that any four vectors oj, , 5, Vi in R are linearly independent.

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Q: Which of the following is the vector projection of (1, 0, 1) onto (-2,4,0)? 1 B. (-1,2,0) C.…

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Q: Determine k so that the vectors (1, –1, k – 1), (2, k, –4), (0, 2, + k, – 8) in R³ linearly…

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Q: Mark each of the following statements true or false:

A: Given,u=4v=5, andu,v=2,   thenu+v=5

Q: Consider the vectors u = (-2, 1, –3) and i = (1, –2, – 1). Find the angle between u and ū. (b) Find…

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Q: Find the vector projection of b = 6î + 3ĵ + 2k onto å = î – 2ĵ – 2k.

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Q: The orthogonal projection of the vector u=(-1,-2) on a=(-2,3) is 8 12 а) 13 13 13 b) 8 13 12 4 c) 13…

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Q: Г10] Let ÿ = 7 and i = -2. Write y as the sum of two orthogonal vectors, j1 parallel to u and j,…

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Q: Prove that the two sets of conditions are equivalent. Here, Tis a linear transformation and u, v are…

A: Given T is a linear transformation and u, v are two vectors and c, d are two scalars. Given…

Q: 4. Which of the following is the vector projection of (1,0, 1) onto (-2, 4, 0)? 2 B. (-1,2,0) c.…

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Q: Let u = and v = and let (u, v) = 2u,v, + 3u,v2 be an inner product. Find a nonzero vector orthogonal…

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Q: The vector projection of u = [3, -4, -2] on the x-axis is: %3D 3 29 3 29 3í

A: The direction ratios of the X-axis are, (1,0,0) Also, u•(1,0,0) = 3 is the scalar projection on the…

Q: Find the projection of the vector a = (2, 5, 1) on the vector b = (1, 1, 3) O a. (10/11)(ax + 3ay +…

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Q: Suppose that u, v, and w are vectors in an inner product space such that (u, v) = 1, (u, w) = 6, (v,…

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Q: Find u × v and show that it is orthogonal to both u and v.u = (−2, 1, 1), v = (4, 2, 0)

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Q: A vector u for which || u || = 1 is called a(n)_______ vector.

A: u is a vector whose modulus is 1. u=1 For a vector, if the modulus or magnitude is 1 then the vector…

Q: 25. Let OP be the position representation of vector A, 0Q be the position representation of vector…

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Q: Find u × v and show that it is orthogonal to both u and v.u = (2, −3, 1), v = (1, −2, 1)

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Q: 1. Consider the vectors (2, 1,0, 2], [3, 1, 0, 1], [1, 1,0, -1] in R'. Decide whether they…

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Q: Find the vector projection of B= 61+ij+2k onto A=1-2j-2k and scalar component of B in the direction…

A: projba = a.ba2b now, <a> = 6i→ + 3j→ + 2k→ <b> = (i→ -2j→-2k→) so, a.b = (6i→+3j→+2k→).…

Q: Let y = #₁ = 1 and = 2 -27 -2 Write y as the sum of two orthogonal vectors, 1 in Span{u} and 2…

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Q: Which of the followings is the vector projection of u onto v where u = -2i + 4j – V5k and v = 2i –…

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Q: 2. Use a directed line segment to represent the vector. (а) и %3D (2, —4) (b) v = (2,4, –4)

A: Given that the vector a u=2,-4  b v=2,4,-4Solution a u=2,-4u=22+-42u=20  =25b v=2,4,-4v=22+42+-42v=6

Q: Suppose that u, v, and w are vectors in an inner product space such that (u, v) = 1, (u, w) = 6, (v,…

A: The norm of the vector, a is defined as:a=a, a. Using the linearity property of the inner product…

Q: 9. Which vector is the orthogonal projection of v = onto W = span 4. 2 V3 V6 V3 V6 O 1 /1\

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Q: Let u = 2i +j- 3k, v = i -j- k. %3D Find the vector projection Projuv o (2, 1, -3) O (1, 1, –3) O…

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Q: Calculate a projection of vector B on A and prophecy of vector A on B. A = 4i - 3j + 2k.     B = 3i…

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Q: Let ū = [1,1,0] and w = [2, 2, 1]. Find the projeci proj,u of the vector w on ū a) [1,0, 0] b) [0,…

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Q: Suppose u, v, and w are vectors in an inner product space such that (u, v) = 1, (v, w) = 0, (u, w) =…

A: Use the values given above for substitution after simplifying the problem.

Q: ) Show that the three vectors vị = (1, 2, 3, 4), v2= (0, 1, 0, – 1), and v3 = (1, 3, 3, 3) form a…

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Q: 5. Show that the vector ortha(b) = b – proja(b) is orthogonal to a. This vector is called an…

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Q: Given the vector u = (4, 1, -6) and a = (1, O, 5). The orthogonal projection of u on a (projau ) is…

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Q: 5. Consider the vectors u = (-2, 1, -3) and i = (1, -2, -1). (a) Find the angle between u and v. (b)…

A: We can use the concept of vector in order to solve the given problem.

Q: . Prove that there is no distinction between contravariant and covariant vectors if the…

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Q: Show that the vector ortha b = b − proja b is orthogonal to a. (It is called an orthogonal…

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Q: Let u = (1, 2, 3, 4), v = (0, 2, t, 3), and w = (1, 1, 0, 1). Find t so that u + v and w are…

A: Here, u=(1,2,3,4),v=(02,t,3) & w=(1,1,0,1)

Q: Find the scalar and vector projections of d = (4, 2, –5) on b = (3, –1, 6).

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Q: Suppose that u, v, and w are vectors in an inner product space such that (u, v) = 1, (u, w) = 5, (v,…

A: According to question,Given data,u, v = 1,u, w = 5,v, w = 0,u=1,v=3w=4

Q: Let ri = xi+yıj+z¡k, r2 = x2i+ Y2j + z2k and r3 = x3i+ Y3j + z3k be the position vectors of points…

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Q: Compute the orthogonal projection of i = 1 onto the line L through 4 and the origin. projz (5) =

A: Vector projection

Q: Find the orthogonal projection of y onto Span{u,, uz} 4 1 1 y =| 2 2 3 7 a. 6 5 3 44 b. 15 37 15 29…

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Q: Show that if C is any two-dimensional vector then it can be written as a linear combination of the…

A: Given: C→ be any two dimensional vector spaceie.,   C→=x,y|x,y∈ℝFormula: Convex set∃ t,s∈ℝ such that…

Q: Find the cross product ax b. a - (1, 1, -1), b= (6, 8, 10) Verify that it is orthogonal to both a…

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Q: Suppose that u and v are vectors in 3-space such that |u||= 5 and u · v = 7.

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Q: Let P = (-2,0, 5), Q = (3, 1, 1), and R = (0,0, 4) and define a = PQ and b= PR. Find %3D (c) The…

A: To obtain the vectors a=PQ→ and b=PR→, find the differences of the points P, Q and P, R,…

Q: 11. Assume u=(1,0,0,-1), v1=(3,2,7,3), v2=(2,1,3,2), and v3=(5,2,9,5). Show that u is in the…

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Q: Let ā = (6, 4, – 10) and b = (2, – 5,6). %3D Compute the projection of a onto b and the vector…

A: We will use basic definitions of the projections and orthogonal component

Q: Suppose that u, u and w are vectors in a real inner product space such that (u, w) = 16, (v, w) = 5,…

A: Instead of u¯,v¯,w¯ , we are writing u,v,w. (for sake of convenience). Given: (i) <u,w>=16(ii)…