The three-dimensional counterpart of a triangle is a tetrahedron, which is a pyramid with four faces. We colour the four faces of such a tetrahedron blue, red, green and purple. Let B, R, G and P be the areas of these faces. In the diagram below, we've also drawn a blue, a red, a green and a purple vector. Let's call these vectors b, r, g, and p. The blue vector is perpendicular to the blue face, its length is equal to the area of the blue face, and the vector is pointing outside the tetrahedron. We have analogous statements for the other three vector/face combinations. (a) Show that b+r+ g+p = 0. Hint. Start by making one of the corners of the tetrahedron the origin 0 = (0,0, 0). Then the tetrahedron can be described by the vectors u = (u1, u2, 43), v = (v1,v2, v3) and w = (w1, w2, w3), as shown in the diagram below. Now express the vectors b, r, g, p in terms of u, v, w and go through the algebra.

Q: If a and b denote the vectors OA and OB, indicate on the same diagram the vectors OC and OD denoted…

A: As first question and second question are independent so I am solving only first question.If you…

Q: |Given the vectors a = (1,2, 3) and b = (1, –1,2), what is the dot product a - b · a? 4 5 14 (7, 1,…

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Q: The dot product of vectors p = (3, -5) and q = (15,9) is A/

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Q: The following are the vertices of a triangle A(4, 0, -1), B(-4, -1, 0), and C(3, 1, 1), find: vector…

A: To find the given values:-

Q: Given the points A: (2,4,-3) and B: (2,-3,5), find the vector ā AB

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Q: Suppose u = (-3, 2) and v = (10, 2) are two vectors that form the sides of a parallelogram. Then the…

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Q: Compute the indicated vectors and also show how the results can be obtained geometrically. b-c

A: Given: b=23, c=-23

Q: If the vectors a=(2,5,-1) and b3(3,-3,k) are orthogonal, the the value of k is

A: Given : a = 2,5,-1b = 3,-3,k

Q: Given points (-1, 5, 3), (3, -2, 4), and (-1, –3, 2). compute the area of the triangle they…

A: * A,B,C are points then unit vector is →AB×→ACII→AB×→ACIIand area oftringle is 12→IAB×→ACI

Q: If two vectors ? = 4? + 6? − 8? and ? = 6? − 8? + 4? are added together describe the characteristics…

A: If two vectors ? = 4i+ 6j- 8k and ? = 6? − 8? + 4? ar

Q: Find the sum of the vectors and illustrate the sum geometrically.u = (4, −2), v = (−2, −3)

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Q: Find the magnitude of the given vector. (2, 6, –2)

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Q: Given the vectors a = (1, 2, 3) and b = (1, –1,2), what is the dot product a · b · a? 4 14 (7,1, –…

A: a=(1,2,3)  and b=(1,-1,2)a.b.a  is dot product of three vectors

Q: 2. The diagram below was created using equilateral triangles congruent to ABSR. If OS=ā and OT=b),…

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Q: Below is a diagram of the dotted cube with edges of length 3. Suppose vectors u = (3,0, 0), v = (0,…

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Q: The following are the vertices of a triangle A(4, 0, -1), B(-4, -1, 0), and C(3, 1, 1), find: vector…

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Q: Find the sum of the vectors and illustrate the sum geometrically.u = (−1, 4), v = (4, −3)

A: Given Data The first vector is u= (-1, 4). The second vector is v= (4, 3).   The expression for the…

Q: 4. ABCD is a parallelogram, and E is the intersection point of the diagonals AC and BD. Name a…

A: It is given that, ABCD is a parallelogram and E is the point of intersection of diagonals. To find:…

Q: If vector a=AB, vector b is BC, and vector c is CD, what is a+b+c using capital letters to write the…

A: Given a=AB→, b=BC,→ and c=CD→

Q: Graph each figure and its image along the given vector. a. AABC with vertices A(2, 6), B(1, 1), and…

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Q: Match the vectors p, q, 7, s, and t with one of the combinations of the u and i vectors from the…

A: Since you have asked a question with multiple parts in a single request we would be answering only…

Q: 2. Find the cross products of the vectors in the diagram above.

A: The concepts to be used: (i) The components of vectors can be written in terms of cosine and sine.…

Q: Find a vector perpendicular to d = [5,–4,6] and h = [-1,3,9].

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Q: 6. Given the following vectors, Draw: a) a-b b) 2c + a c) - (c-b) a X b of

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Q: Given the vectors à = and b =

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Q: Which of the following vectors represents - 2u + v?

A: Given,

Q: Find three different vectors that are in the span of the given vectors. 9- 7 U2 = 4 -2 6. 2

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Q: Is the following statement correct? If A, B, C, and D are distinct nonzero vectors represented by…

A: A, B, C, and D are distinct nonzero vectors represented by arrows from the origin to the points A,…

Q: Find the length of the vector. v= (4, 3) M -| 43

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Q: In Figure , A, B, C, D,E , and F are the vertices of a regular hexagon centered at the origin.…

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Q: Find the unit vector with the opposite direction of a⃗=(8,−10,3,−1,9,3)

A: We have to find the unit vector with the opposite direction of a→=8,-10,3,-1,9,3.

Q: Given three vectors A, B, and C as follows, A = a, + a,2 – a 3, В 3 — а,4 + а,, С 3D а,5 — а,2, find…

A: Please see the explanation below

Q: Suppose that A, B, and C are vertices of a triangle and that a, b, and c are, respectively, the…

A: Let A=(x1,y1,z1)B=(x2,y2,z2)C=(x3,y3,z3)

Q: Match the vectors p,7,ï, 3, and t with one of the combinations of the i and i vectors from the…

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Q: The length of the vector c = (2, -3, 1) is %3D

A: Length of vector equal to magnitude of that vector  if v = <x,y,z>  then magnitude/Length =…

Q: Given three vectors A, B, and C as follows, A = a, + a,2 – a,3, В %3D — а,4 + а,, С 3D а,5 — а,2,…

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Q: The sum of two vectors is called the ___________, and it can be represented geo-metrically as the…

A: The sum of two vectors is called the resultant, and it can be represented geometrically as the…

Q: 10î - 4ĵ +7 k̂ and B =

A: Find the dot product A∙ B and its angle of the given vectors: A = 10î - 4ĵ +7 k̂ and B = -2î +ĵ +…

Q: ) Below is a diagram of the dotted cube with edges of length 5. Suppose (5, 0,0), v = (0, 5, 0), and…

A: Given u=5,0,0v=0,5,0w=0,0,5

Q: Given points: A = (4, 9, 1) B = (-2, 6, 3) C = (6, 3, -2) Using norm of located vectors, prove that…

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Q: In Figure , A, B, C, D,E , and F are the vertices of a regular hexagon centered at the origin.…

A: a=OA→  &  b=OB→

Q: Given the three vectors A, B, and D, show that A (B +D) = (A · B) + (A · D).

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Q: If two vectors have the same magnitude, then they are equal.

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Q: Graph the vectors , , and below. The vectors may start at any point. 5+ 4- 3 -5 -4 -3 -2 -1 3. -1…

A: The answer is found in step-2

Q: Below is a diagram of the dotted cube with edges of leng Suppose vectors u = (3,0, 0), v = w =…

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Q: In the following diagram, the rectangle ACEG is divided into four congruent rectangles. F E a) List…

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Q: Match the vectors p, q, r, s, and t with one of the combinations of the u and v vectors from the…

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Q: If ả = 2ỉ + 33, b = j – ỉ and ở = 33 are vectors in 2D, then determine the of a + 3b + c

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Magnitude

A vector is an element with both direction and magnitude. A vector is represented using an arrow, where the arrow represents the direction of the vector and its length is represented using the magnitude. The set of vectors in space is known as vector space. The vectors can be added, subtracted, multiplied, and divided in a vector space. The image below shows a vector AB.