Prove that if the vector x emanates from the origin of the Euclidean plane and terminates at the point with coordinates (a1, a2), then the vector tx that emanates from the origin terminates at the point with coordinates (ta1, ta2).

Q: Prove Ảx (B x Č) = B(Ả C) – Č(Ã B) is true by expanding Ả, B, and components (or Cartesian…

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Q: Let u and v be non-zero vectors in 3-space. Prove that u and v are perpendicular if and only if |u…

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Q: Find a non-zero vector 7 perpendicular to the vector u 12 [3]. -6

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Q: In Cartesian space, the direction of the line (x, y, z) = (a, b, c) +r(d, e, f) is represented by…

A: The given line is (x,y,z)=(a,b,c)+r(d,e,f) So, the point (a,b,c) is on the line and (d,e,f) is a…

Q: Prove that if the vector x emanates from the origin of the Euclidean plane and terminates at the…

A: The given  vector x→ emanates from the origin and terminates at a1, a2: x → = a1, a2 - 0,0 = a1,…

Q: 4

A: We are given the following vectors.

Q: Express the vector from P(-1, -4, 6) to Q(1, 3, -6) as a position vector in terms of i, j, and k.

A: Given vectors are P-1,-4,6 and Q1,3,-6.

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

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Q: Find two vectors of norm 1 that are orthogonal to the vectors u = and v =

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Q: Find the projection of vector A=(6,7) along vector B=(1,4).

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Q: What are the coordinates of the vector 0 in the Euclidean plane that satisfies condition - "There…

A: The co-ordinates of the 0 vector in the Euclidean plane is

Q: Find the projection of vector w = onto vector v = . Round all components to the nearest hundredth.

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Q: Prove that for any vectors u, v, w , the set span{u+v,u – v, 3u – 3v + 2w} contains u, v, and w .…

A: A vector v is said to be contained in span of set S  (span(S)) if vector v can be written as a…

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

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Q: Consider the set V of all real triples of the form (a, b, c) where c = a - 2b under the usual vector…

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Q: Find a vector in the plane parallel to the line ax + by = c.

A: We know that, Slope of parallel lines are equal Given equation of line is ax + by = c Slope of this…

Q: Find the length of the vector. v = (-2, 0, 5, 1) %3D ||| M = %3D

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Q: Compute the distance between the vectors u = and

A: Given vectors are u=42−16 and v=4315. We have to find the distance between the vectors u→ and v→.…

Q: Let x and y be two orthogonal vectors. Then prove that +

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

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Q: Show that v, = v 2 V2 3 span R and express the vector v= as a linear -7 %3D combination of v,,v2.

A: Solve the following

Q: Show that the vector v = ai + bj is perpendicular to the line

A: GivenVector V = ai + bj and the line ax + by + c

Q: Use vectors to prove that the line connecting the midpoints of two sides of a triangle is parallel…

A: Let the arbitrary triangle has vertices A, B and C has three vectors AB, →AC →  and  BC → . Let M…

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: . Prove that (only one): E - 4y-V3ay is a unit vector? a- F 2 b- A = 6ax - 3a, + 2a, and B = 3a, +…

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

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Q: Let A, B, C be points in 3-space. Suppose that AB represents u, BC represents v, AC represents W.…

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Q: | Show that the vectors 2 do not span R³ by giving a vector not in their span.

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Q: Show that the vectors do not span R' by giving a vector not in their span.

A: Solution :-

Q: Prove that the points position vectors collinear. With at 26, 2a-5 and 3a-46 are

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Q: 9. Show that for any real numbers a and b the vectors u = ai and v = bj are orthogonal?

A: Topic:-  vector algebra

Q: If vector A is any non-zero vector, a unit vector in the same direction of A would be?

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Q: Find two vectors in opposite directions that are orthogonal to the vector        u = ⟨3, 1, −2⟩

A: u=3, 1, -2 Let v=a, b, c be the vector orthogonal to the vector u=3, 1, -2. Then, u·v=0…

Q: Find a unit vector that is orthogonal to both ı̂ +ȷ̂  and ı̂ +?̂

A: Given query is to find the unit orthogonal vector to the given vectors.

Q: Show that if A is both diagomalizable and invertible, then so is A-1. Find a unit vector in the…

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Q: . Let x and y be two orthogonal vectors. Then prove that

A: If x is any vector then Euclidean norm or 2-norm is denoted by ||x||2 and is given by  ||x||2 =…

Q: Use vectors to prove that the line joining the midpoints of two sides of a triangle is parallel to…

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Q: Show that if u + vand ủ – v are orthogonal, then the vectors u and must have the same length

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Q: Let b = 4i + 3j and c be two %3D vectors perpendicular to each other in the xy-plane. All vectors in…

A: Given: b→=4i+3j and c→ are two vectors perpendicular to each other in xy-plane.  To find: All the…

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: Determine whether the vector v = 8 is spanned by the vectors and 3 or not.

A: Given three vectors and we have to find if  v=789 is spanned by111 and 123

Q: Find a unit vector u that had the same directions as the vector v= (-3,3,-2)

A: Use the basic property of unit vector

Q: Show that the vectors ( 1, 0,1) ; ( -1, 2,3) and (0, 1; -1) spans R³

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Q: Find three vectors v1, v2, v3 such that (1) any two span a plane, and (2) Span{v1, v2, v3} is also a…

A: Span of two vectors is a plane

Q: Using the vectors from the previous problem, find the projection ul of v onto the vector u and find…

A: We say that 2 vectors are orthogonal if they are perpendicular to each other. i.e. the dot product…

Q: 64. Show that if u + v and u – v are orthogonal, then the vectors u and v must have the same length.…

A: here, the given vectors, A =u+vB=u-v given that, they are orthogonal, that is, A·B=0u+v·u-v=0⋯⋯1

Q: Find a so that the vectors v = i- aj and w = 7i + 7j are orthogonal. ..... une on integer or o…

A: If two vector orthogonal then dot products will be zero

Coordinate Geometry

Coordinate geometry, also known as analytic geometry or Cartesian geometry in classical mathematics, is a type of geometry that is studied using a coordinate system. Coordinate geometry is a branch of mathematics that describes the position of points on a plane using an ordered pair of numbers called coordinates.