Consider the two vectors A = Axî + Ayĵ and B = Bxî + Byĵ. If î · î = ĵ • ĵ = 1 and î · j = ĵ · î = 0, prove algebraically that à · B = A,Bx + A,By.

Q: Let V be the set of vectors shown below. V= x> 0, y>0 a. If u and v are in V, is u +v in V? Why? b.…

A: Please check step 2 for solution.

Q: If u is a unit vector and a is a positive scalar satisfying au + = 0, find a and u.

A: If u is a vector and a is positive scalar satisfying   au+1,-2,5=0.  To find : a and u.

Q: Prove that for any non-zero vectors a, b, a · b = |a||b| cos(0), where 0 is an angel between a and…

A: Consider the n -dimensional vectors a and b , where n≥2 . The vectors a , b and a−b can be…

Q: Let A and B be nonzero vectors. Find k e R such that A – kB is orthogonal to A + B.

A: We know if a vector P→ is orthogonal to another vector Q→ then scalar product (dot product) is zero.…

Q: 1. Given the vectors U = i- 2j+k, V = 2i+j+k and W = 3i – j+ 2k, compute the following: U (c) ||U||

A: The solution is :

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: Assume that the vectors u = ai-j+k and w=i+9j+5k orthogonal. Then the are value of a is

A: If two vectors are orthogonal (perpendicular), then their dot product is 0. We use this to solve the…

Q: Find projwv. Then decompose v into two vectors, v1 and v2, where v1 is parallel to w and v2 is…

A: v=2i+4j  and  w=-3i+6j. Now, v·w=2i+4j·-3i+6j=2-3+46=-6+24=18 & w=-32+62=45=35 Therefore,…

Q: For any two nonparallel and non orthogonal vectors a and b with angle between them, it holds that…

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Q: Given the vectors U =i- 2j +k, V = 2i +j+k and W = 3i - j+2k, compute the following: (a) Ux V (b)…

A: We will apply dot product formula and cross product formula.

Q: Consider the two vectors A = A,i + Ayj and B = B,î + Byj. In cross product, %3D %3D ixî = j xj = 0,…

A: We have given two vectors A→=Axi^+Ayj^  and  B→=Bxi^+Byj^ i^×i^=j^×j^=0i^×j^=k^, j^×i^=-k^   Find…

Q: Consider two vectors A and B prove that the cross product between them can be calculated as follows…

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Q: (d) Suppose the three vectors x, y, z E R³ are known to be linearly independent. Are the three…

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Q: 1. Given the vectors U = i – 2j+k, V = 2i+j+k and W = 3i – j+2k, compute the following: (a) U. V (b)…

A: Given vectors   u = i^ - 2j^ + k^  ,  v = 2i^ + j^ + k^and w = 3i^ - j^ + 2k^We find a u.v b u×v c…

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 V be the set of vectors shown below. V= x<0, y< a. If u and v are in V, is u + v in V? Why? b.…

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Q: Consider two vectors →A=Ax^i+Ay^j+Az^k and →B=Bx^i+By^j+Bz^k. Scalar product of these two vectors…

A: The scalar product can be evaluated using the two formulas given in the question. We will take the…

Q: Let a = 2i – j - 3k and b = 3i – j-k %3D (a) C ) Find two unit vectors orthogonal to both a – i and…

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Q: Given v = 2i + j, w = 6i + 3j, find projw v. Then decompose v into two vectors, v1 and v2, where v1…

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Q: Find projwv. Then decompose v into two vectors, v1 and v2, where v1 is parallel to w and v2 is…

A: To Determine: Find projwv. Then decompose v into two vectors, v1 and v2, where v1 is parallel to w…

Q: Find projwv. Then decompose v into two vectors, v1 and v2, where v1 is parallel to w and v2 is…

A: we have to find projwv of the given vectors and then we have to decompose v into two vectors, v1 and…

Q: 14. Find a vector perpendicular to both of the vectors * (ũ + v) and (ũ – v) where ủ = 2i – j + k,…

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Q: Q. For the Spheri cal polas Coozdinates (r0,d), prove that their unit vectors mutually orthogonal?…

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Q: If the triple scalar product of the vectors A = i+ 3j + k, B = 2i -j-k and C= ai + 7j + 3k is zero ,…

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Q: If a = i+ 2j + k and b = i+ 5j +k, find a unit vector with positive first coordinate orthogonal to…

A: Given that     a = i+2j+k and b = i+5j+k We have to find a unit vector with positive first…

Q: Find projwv. Then decompose v into two vectors, v1 and v2, where v1 is parallel to w and v2 is…

A: we have to find projwv. Then decompose v into two vectors, v1 and v2, where v1 is parallel to w and…

Q: Determine whether the vectors u= i + 5j − 2k,  v = 3i − j and w = 5i + 9j − 4k are coplanar

A: We have to determine whether the vectors u= i + 5j − 2k,  v = 3i − j and w = 5i + 9j − 4k are…

Q: Prove that two vectors v = (a, b) and w = (c, d) are perpendicular if and only if ac + bd = 0

A: we have to prove that the two vectors v=a,b and w=c,d are perpendicular if and only if ac+bd=0

Q: Use the vectors u = a1i + b1j, v = a2i + b2j, and w = a3i + b3j, to prove the given property : u . v…

A: Consider the following vectors: u→=a1i^+b1j^v→=a2i^+b2j^…

Q: If the vectors b, c, d are not coplanar, then can we say that the vector (ā×b)×(ē×d) + (ā×ē) × (dxb)…

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Q: Find a set of vectors {u, v} in Rª that spans the solution set of the equations w – x + 2y – 4z 0,…

A: The detailed solution is given below Thank you

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: If a and b are vectors with the same initial point and angle between them, prove that||a - b||2 =…

A: Consider the given expression as, a-b2

Q: The angle between the unit vectors a and b is 0. By expanding the scalar product, show that (3a –…

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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  &amp;  w=-2i+6j Now, v·w=2-2+46=20 &amp; w=-22+62=40 Now,…

Q: Le and ü .Write j as the sum of two orthogonal vectors, a, in Span {u} and i2 orthogonal to ū.

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Q: Consider two vectors A and B prove that the cross product between them can be calculated as follows…

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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. Find two unit vectors orthogonal to both j − k and i + j

A: We will find the two unit vectors orthogonal to j-k and I+j as following.

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

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Q: Prove that |A|B + |B|A is orthogonal to |A|B – |B|A, for any vectors A and B.

A: We need to show that |A|B + |B|A is orthogonal to |A|B – |B|A, for any vectors A and B. Here, |A|…

Q: b) Find out the Vector =y²z?i+x?z?j+z²y²k is irrotational or not?

A: By definition, Any vector is said to be irrotational if the curl of the vector is zero. The given…

Q: IH A and B are any two vectors in V, prove that (A- B) x (A + B) = 2(A X B).

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Q: If a = i+ 3j + k and b = i+9j + k, find a unit vector with positive first coordinate orthogonal to…

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Q: Given v = 6i + 2j and w =i-j, a. Find projwv. b. Decompose v into two vectors v, and V2, where v, is…

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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.…

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Q: s such that vectors –si – 2j+5sk and 9si +j+3sk are orthogonal.

A: