(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 w) × v = 0.

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: Let v, w, and a be nonzero vectors such that v · a = w · a.Is it true that v = w? Either prove this…

A: Given: v, w and a is non zero vectors, such that: v·a=w·a To prove: v=w

Q: Suppose that the vectors {v, u, w} are linearly independent in some vector space V. Prove that the…

A: Given that the vectors u, v, w are linearly independent in some vector space V. Then the linear…

Q: Verify the Pythagorean Theorem for the vectors u and v.u = (1, −1), v = (1, 1)

A:

Q: Suppose u, v and w are nonzero vectors in 3-space and a is a scalar. Which of the following are…

A: We have to choose the correct answer from the given options

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

A:

Q: Suppose u, v and w are nonzero vectors in 3-space and a is a scalar. Which of the following are…

A:

Q: et u (x, y) E R² any vector, prove that l|u||

A:

Q: 4) Suppose x and y are vectors in R". Prove a) r+ >f -x->f =4(x• y) b) If x and y are orthogonal…

A:

Q: Let v = 2i - 5j and w = i - j.Decompose v into two vectors, v1 and v2, where v1 is parallel to w and…

A: v=2i-5j  and  w=i-j. Now, v·w=2i-5j·i-j=21+-5-1=2+5=7 & w=12+-12=2

Q: Let f : R² → R be defined by S if (x, y) ± (0,0), 1³y? f(x,y) = if (x, y) = (0,0). Determine the…

A: According to the given information, it is required to determine a non-zero vector u = (h,k) for…

Q: 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 w)…

A:

Q: Let ƒ : R² → R be defined by A if (x, y) # (0,0), f(r, y) = if (r, y) = (0,0). Determine the nonzero…

A: According to the given information, it is required to determine a non-zero vector u = (h,k) for…

Q: 4. Prove that, for any vectors A, B and C, that a necessary and sufficient condition А x (Вx С) %3…

A: The solution is:

Q: Find a vector B such that F = V × B and V · B = 0, where F %3D (x + Зу, у — 2z, х - 2z).

A:

Q: For any vectors u, v and w, show that uxv=uxw does not always imply v = w. Prove that u v ≤ ulv, and…

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Q: Show that the vectors u=, v= , and w= satisfy the following properties of the cross product. и хи30…

A:

Q: Let u = (−1, 1), v = (0, 1),and w = (3, 4). Find the vector x that satisfies u−2x = x−w + 3v.

A: u=-1, 1,  v=0, 1  &  w=3, 4

Q: Prove that if vector u is orthogonal to both vectors v and w, then u is orthogonal to v + w.

A:

Q: Make all obvious simplifications.

A: We have to solve

Q: Prove that ||v×w||2 +|v.w|2=||v||2||w||2 for all vectors v,w in R3

A:

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

A:

Q: Show that 0 x a =0 = a x 0 for any vector a in V3

A: lets take vectorv=(x,y,z) and   0=(0,0,0)

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: Let v = 2i - 5j and w = i - j. Decompose v into two vectors, v1 and v2, where v1 is parallel to w…

A: Given vectors are: v = 2i - 5j and w = i - j The formula for the vector projection is given by…

Q: Let u = (-1,0,1), v= (3,2, 1) and w = (0,1, 2) . (a) Prove that the vectors u – v, u+ v and u x v…

A: Given : u=<-1,0,1> v=<3,2,1> w=<0,1,2>

Q: Prove that two non-zero vectors u and v in n-space are perpendicular if and only if u • v = 0

A:

Q: (1) Let x and y be the vectors x = and y = 4 (la) Find a vector z such that ||z|| = 1 and z is…

A:

Q: Let v = 2i + 4j and w = -2i + 6j. Decompose v into two vectors, v1 and v2, where v1 is parallel to w…

A: v=2i+4j  &  w=-2i+6j Now, v·w=2-2+46=20 & w=-22+62=40 Now,…

Q: prove that 12 div A; = √√√x² {√8 8¹k Ax} = dxr div A¹, where A; and Ai are the covariant and…

A:

Q: Let u= and w- Find the unique vector v in R'such that u.v 1, v is orthogonal to w, l|v|| V3, and y>…

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Q: The vector w is a unit vector, and the vector i is any non-zero vector. If z = (x · W)w and ỹ = i –…

A: Use the property of dot product and associative and distribute property to solve the problem.

Q: Prove that for any vectors u, v, and w in R2 or R3, we have u ·(v + w) = u · v + u · w.

A: Let u=u1, u2, u3  ,  v=v1, v2, v3  &  w=w1, w2, w3 be the vectors. Now, v+w=v1, v2, v3+w1, w2,…

Q: Prove that for any vectors u, v E C", and scalar z € C, a) u· v = v•u b) zu v = z(u · v)

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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,…

A:

Q: Suppose that u, v are vectors in R³, u · v = 1, || = 1, the angle between u and v is T/3, and u is…

A: According to the problem, we have

Q: For any vectors U, V and scalars a, b, prove that (a) (b) (aU) × (bV) = ab(U × V) (aU) · (bV) = ab(U…

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Q: 3. For any vectors U, V and scalars a, b, prove that (a) (aU) x (bV) = ab(U x V)

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Q: For any vectors U, V and scalars a, b, prove that (aU) x (6V) = ab(U × V) (a) (b) (aU) - (bV) = ab(U…

A:

Q: 10. Let the vector v = 2i -3j + k, and the vector w = = -i+3j+ 2k. Find v x w.

A:

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

A: Condition of dot product.

Q: If in R a vector u is orthogonal to vectors v and w, then { u, v, w} is an orthogonal set. True…

A:

Q: Let x and y be any two vectors in R^ 3 . a. Prove that the vectors ||y||x + ||x||y and ||y||x -…

A:

Q: Linear algebra

A: From the given information:Let,

Q: Let W = {[x]∈ R2^ | y > or equal to 0} .             [y] (a) If u and v are vectors in W , is u + v…

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

A:

Q: For any two vectors u and v, prove that |u × v|2 + (u · v)2 = |u|2 |v|2

A: Here u and v are two vectors ,   We have to show   |u × v|2  + |u · v|2 = |u|2 |v|2

Q: Consider the collection of vectors {v1, v2, V3}, where v1 = -3v2 + 7v3. Is the collection linearly…

A:

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: