Determine whether the following statements are true or false. Prove if it is true or give counter example if it is false. (a.) If u and v are vectors in R" such that their length are 1 and the inner product u· v = =7, then the distance between them is 3.

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: Find a unit vector that is orthogonal to both u = i − 4j + k and v = 2i + 3j.

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Q: Find a unit vector orthogonal to both u and v. u = 8i – 16j + 2k v = 16i + 32j – 6k -

A: In the given question we have to find the unit vector orthogonal to both u and v. u=8i-16j+2k…

Q: Use the dot product to determine whether the vectors are parallel, orthogonal, or neither. v= 2j. w=…

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Q: 3. Use the dot product to determine whether the vectors are parallel, orthogonal, or neither. v = 4i…

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Q: State whether the vectors are parallel, orthogonal, or neither. 5) v = 2i + j, w=i- 2j

A: As per our guidelines we are supposed to answer only one question.Kindly repost other question as…

Q: snip

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Q: Find the Euclidean inner product of the given vectors. 3 4 u = v = (u, v) = Ex: 5

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Q: Given the vectors in parts a) and b), determine whether u, v, and w lie in the same plane when…

A: We have to determine whether vectors u, v and w lie in the same plane when positioned so that their…

Q: Find a vector v such that v = 15 and v is in the same direction as –li + 13j

A: The solution is given as

Q: Decompose v into two vectors, v1 and V2, where v is parallel to w and v2 is orthogonal to w. v=i+j,…

A: A vector parallel to w can be obtained from the following formula, v1=v.ww2w and the other vector…

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: Prove the stated property of distance between vectors. d(u, w) <= d(u, v) + d(v, w) for all…

A: We have to prove the property of distance between vectors as,

Q: For the vectors A=x + 2ŷ – î = (1, 2, – 1), B=ŷ + î = (0, 1, 1), C=î-ŷ = (0, 1, 1), a) Prove that Ax…

A: Correct last vector coordinates representation

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 the vectors 'v (1, 0, 1) and w (2,0, -2). %3D (a) Are 'v and w perpendicular? (b) Show that…

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Q: Prove for generic vectors u, v, w in R3 and c, d each vector property can be reduced to proper…

A: Addition of Vector: The term "addition of vectors" refers to the joining of two or more vectors.…

Q: Prove that if the vectors a,b parallel, then āxb=0. are %3|

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Q: Find the vector z, given that u = , v = , and w = . z = 8u - 3V - -W 2 1. Z =

A: We have been given the vectors  u = &lt;1, 2, 3&gt; v = &lt;2, 2, -1&gt; w = &lt;4, 0, -4&gt;

Q: Decompose v into two vectors,one parallel to w and the other orthogonal to w. v = 2i + j; w =4i + 3j

A: we have to decompose v into two vectors, one parallel to w and the other orthogonal to w. v = 2i +…

Q: Show that if the vectors v, w are orthogonal, then ||v + w||^2 = ||v||^2 + ||w||^2.

A: Given: The vectors v, w are orthogonal, then ||v + w||^2 = ||v||^2 + ||w||^2.

Q: Find b so that the vectors v = i + jand w = i + bj are orthogonal.

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Q: Decompose v into two vectors v, and v2, where v, is parallel to w and v, is orthogonal to w. v = 2i…

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Q: Consider the vectors u = (2,0) and v = (-4,4). Sketch the vectors, find the angle between the…

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Q: Prove that the vectors v= and w= are orthogonal.

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Q: Determine a unit vector perpendicular to both A=i+2j-3k and B=-2i-3j+k

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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: Let a, b, c and y be the three dimensional vectors à = 3j+4k, b=i+6j+2k, č=4i-8j, ÿ=−i+ Perform the…

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Q: Let u = a1i + b1 j, v = a2 i + b2 j, w = a3 i + b3 j. Prove the given property by obtaining the…

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Q: Find the cosine of the angle between the vectors P1 (x) = 1 + 2xr – x² and p2 (x) = -2 – 3x + 2x² -…

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Q: 16. Consider this regular octagon. If AB =d and BC =v, express each %3D vector in terms of u and v.…

A: Considering OD on x-axis. Angle at center O made by any side of regular octagon is 45degrees.

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: The vectors a and bare unit vectors such that a a+b and 2a +jb are perpendicular. --Determine the…

A: The given vectors a→ and b→ are unit vectors such that a→.b→=12.Given: a→+b→ and 2a→+jb→ are…

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: 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: Prove the stated property of distance between vectors.d(u, w) &lt; d(u, v) + d(v, w) for all vectors…

A: Basic logic and properties of vectors can be used to solve the question.

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: If a and b are both unit vectors, then axb is a unit vector only when a and b are…

A: since you have asked multiple questions in a single request, we would be answering only the first…

Q: Find the values of ( x ) such that the vectors and are orthogonal.

A: Given: The two vectors are: 3,2,x and 2x,4,x Also, the two vectors are orthogonal.

Q: Find a unit vector that is orthogonal to both u and v.       u = −3i + 2j − 5k,    v = i − j + 4k

A: u=-3i+2j-5k,  v=i-j+4k Now the vector orthogonal to both u and v is, w=u×v=…

Q: Determine whether each of the following statements is true or false. You do NOT need to explain. Let…

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Q: Given nonzero vectors u, v, and w, use dot product and crossproduct notation, as appropriate, to…

A: It is known that the unit vector along the direction v is given by vv. Since it is an unit vector,…

Q: Find a vector B for which that is parallel to A =-6i +3j –2k but has the opposite direction.

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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: Let u=i-3j+2k, v=i+j, and w=2i+2j-4k be vectors in 3-space. Find: (a) ||u+v|| (b) unit vector in the…

A: given vectors u=i-3 j+2 kv=i+jw=2 i+2 j-4 k find (a) u+v (b) unit vector in direction of vector w…