Let u and v be u = [3,0,3] and v = [1,1,4]. Verify that: |u x vl? = |u]?\v]? – (u · v)²

Q: Let t be a fixed real number and let c = cos t, s = sin t, x = (c, cs, cs2, . . . , csn−1, sn)T Show…

A: If x is a unit vector then:

Q: Prove that Cov (x, y) = E (xy) – E (x) E (y).

A: Cov(x,y) is covariance of x and y. E(x) is expected value of x E(y) is expected value of y.

Q: find the distance d(u, v) between u and v in the given exercise.

A: We will use the formula

Q: Find the length and direction (when defined) of u x v and vxu. u= 5i + 3j – 5k, v=0

A: we know that  uxv=|u|.|v|sin∅.n^where ∅ is angle between u and v and n^ is unit vector perpendicular…

Q: Let u, EC prove that ū · v = v · ū น

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Q: et T E L(V,W), and U e L(W, X). Prove that UT : V → X is also linear

A:

Q: find the distance d(u, v) between u and v in the given exercise.

A:

Q: Prove that u is orthogonal to v - proju(v) for all vectors u and v in Rn, where u ≠ 0.

A:

Q: Show that u and v are parallel if and only if u×v = 0.

A: The cross product of two vectors are defined as u×v=uvsinθ. Two vectors are parallel if the angle…

Q: For all u and v in R" with u + 0, show that u is orthogonal to v – u•v

A:

Q: Show that: (a) Cov(X,Y) = E[XY] – E[X]E[Y]. (b) If X and Y are independent, then fY|X(y|x) = fY(y)

A: Solution:a.Show that COV(X,Y)=E(XY)–E(X)E(Y)

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

A:

Q: Compute u x v. (a) u = 2i + 3j+ 4k, v = -i+ 3j – k (b) u =i+ k, v = 2i + 3j – k (c) u =i-j+ 2k, v =…

A: Note: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts…

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

A: Prove the following

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

A:

Q: Consider the following. (4,-1, 0) 7, 0) Find u x V. Determine if uxV Is orthogonal to both u and v…

A: We have given  u=⟨4,-1,0⟩v=⟨-4,7,0⟩ Then, u×v=ijk4-10-470=i0-0-j0-0+k28-4=24k

Q: Prove that u × v is orthogonal to both u and v.

A: Consider the two vectors u and v, Evaluate the cross product of the two vectors u and v,

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

A: Given vectors

Q: Prove that da(x) = a(b) a(a), %3D |

A: This is a problem from Reimann Stieltzes Integral

Q: Consider the following. u = (-2, 2), v = (1, 1) %3D %3D (a) Find u v.

A: Given two vectors:

Q: Prove that f, uv v • dỉ = – §̟ vũu • dĩ

A: Whenever the force is conservative in nature the work done over the closed loop is always zero. i.e…

Q: Let u = (1, 3), v = (2, 1), w = (4,−1). Find the vector x that satisfies 2u −v + x = 7x + w.

A:

Q: A set of vectors {u1,... , Um} in R" is called orthonorma 1 if i = j 0 if i + j. 1 %3D U¡ · Uj

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Find r' (2), where r(t) = u(t) × v(t), with u(2) = (1, ,2, – 1), u' (2) = (3, ,0, 4) , and v(t) =…

A: Solution :-

Q: Find u × v and show that it is orthogonal to both u and v.u = (4, 1, 0), v = (3, 2, −2)

A: Now cross product is,

Q: Find the distance between u and v.u = (1, −1), v = (−1, 1)

A:

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: Use Cramer's rule to solve for y without solving for x, z, and w. 5 x + y + z + W = -7 5x + 3y - z +…

A: Here we have to use cramer rule for solving the system of linear equations

Q: 2. -()-- () Which of the following vectors X belongs to the linear envelope [u, v], where u =1 and v…

A: vectors x belongs to the linear envelope [u, v] only if we have x = au+bv for some real values a and…

Q: Find (u, v), ||ul|, |v||, and d(u, v) for the given inner product defined on RT. u = (1, 3, 0), v =…

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Q: For (a, B) E R?, define a function (-, -) : R² × R? → R by setting arr' + xy + Yx' + Byy' . 1. Find…

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Q: For x, y, z, w > 0, prove that (x/z)+(y/z)+(z/w)+(w/x)≥4

A: It should by x/y, instead of x/z

Q: Show that fi(r) = rª and f2(r) = r² +1 are orthogonal on the interval [–1, 1).

A:

Q: For which value of x do the following vectors form a linearly dependent set in R V =| x, x,

A:

Q: Find (u, v) for (u, v) = 2u,v, +3u, v, +u,v, defined in R, where u (9,0,-3) and v (8, 2,17). A. (u,…

A:

Q: Let r E R and --(), v- (), -- (1). 3 u = V = W = 2 For which values r ER is the set {u, v, w}…

A:

Q: If w = f(u, v), u = x + y, and v = x - y, show that w,wy = w - w?. %3D %3D

A:

Q: Suppose u, VE Vand ||u|| < 1 and ||v|| < 1 . Prove it: V1- ||u||?/1- ||u||2 <1- (u, v)|

A: Suppose u, v∈V and u≤1 and v≤1 We have to prove that 1-u21-v2≤1-u,v Here, we have  u≤1 and v≤1 Also,…

Q: Find the length and direction (when defined) of u x v. u = -,i + +k U = i + j + 2k ||uxo ||= 8 | ||u…

A:

Q: Prove that u, v, w ∈ R3     and [ u x v, v x w, w x u] = [ u, v, w]2

A:

Q: Suppose u, v E L. Prove that ((u ^ (¬v)) V v = u v ((¬u) A v)). a.

A:

Q: In Rn with inner product{x, y} = xTyderive a formula for the distance between two vectorsx = (x1, .…

A: Take two vectors x and y in an inner product space x=x1,…,xn and y=y1,…,yn then x,y=xTy

Q: Is it true that if x is orthogonal to v and w, then x is also orthogonal to v − w?

A: x is orthogonal to v and w. Then their dot product is zero. That is, ⇒x·v=0  &amp;  x·w=0

Q: Let w = f(u), where u = x g(y), show that w,xx = f" (x g(y)). g²(y). %3D

A: Let w=f(u), where  u=x g(y). So using the chain rule, differentiating with respect to x, we get…

Q: Find a and b such that v = au + bw, where where u = (1, 2) and w = (1, –1). v= (0, 24) a = %3D

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Q: Find u × v and show that it is orthogonal to both u and v.u = (12, −3, 1), v = (−2, 5, 1)

A: Given vectors u=12,-3,1v=-2,5,1 Their cross product is u×v=ijk12-31-251…

Q: If i • ( or v• (u x w) or w • (u x 7) equal to zero, then the 3, and w vectors u. are coplanar.…

A:

Q: Suppose v1, V2, ..., Vm E V. Prove that {vı, V2, ..., Um}+ = (span(v1, v2,….., Vm))“.

A: Given,v1,v2....vm∈V To prove :      v1,v2....vm⊥=span(v1,v2....vm)⊥⇒v1,v2....vm  is a set of…