Consider some perpendicular unit vectors vi, v2, .. , Vm in R". Show that these vectors are necessarily linearly independent. Hint: Form the dot product of vi and both sides of the equation CU1 + C2 U2 +.. + C;V; +.. + Cm Um

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: Prove the stated property of distance between vectors. d(u, v) = d(v, u) for all vectors u and v

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Q: If r1,r2 and r3, are three parallel vectors of the same magnitude, Prove that r1+r2+r3 is equally…

A: GIVEN: r1 , r2 , r3 are three parallel vectors

Q: Find the value of a so that vectors U = ai + 3j and V = 9i + 9j are perpendicular. a =

A: Given: U=ai+3jV=9i+9jθ=90ο

Q: Prove the stated property of distance between vectors.d(u, v) = d(v, u) for all vectors u and v

A: Given the stated property d(u, v) = d(v, u). We need to prove the above stated property.

Q: Determine the two unit vectors that are orthogonal to i = (1,2). Which two unit vectors are parallel…

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Q: If A : li + 4j – 8k, B = -6i + 5j + 6k, C 9, and C is perpendicular to both A and B, write the C in…

A: Given, A→=i^+4j^-8k^B→=-6i^+5j^+6k^ Let, the required C→ vector be, C→=c1i^+c2j^+c3k^.......(1)

Q: Let u =j- 3k, v = - 9i + 3j – 3k, w = 3i -j+ k. Which vectors, if any are (a) perpendicular? (b)…

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Q: In general, show that the vectors V = ai + bj and W = -bi +aj are always perpen-dicular. Assume a…

A: Given vectors are v=ai+bj and w=-bi+aj, where a≠0 ,b≠0. We know that two vectors are perpendicular…

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: Let a and b be two vectors such that Jal =17, b is a unit vector, and 0= " is the angle between…

A: Solve for

Q: Rewrite the following vectors in term of unit vectors а. ОP %3D (9, -2) b. 0Q = (7,0,8)

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Q: What is the area of the parallelogram formed by the vectors v = (1,0, 1) and w = (0, 2, 2) tail to…

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

A: Hence the two vector u and v have same length.

Q: Find the value of a so that vectors U = ai + 9j and V 3i + 12j are perpendicular. %D a =

A: Since, We know that Two Vectors are perpendicular to each other if and only if their dot product is…

Q: Use the dot product to find the angle between the vectors v =i – 2j + k and w = i+j-2k. For which…

A: First we have to find the angle between the vectors v and w where v=i-2j+k and w=i+j-2k. For this,…

Q: If 7, r, and r, are three parallel vectors of the same magnitude Prove that i+r, +r, is equally…

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Q: A. Give the Cross products of the unit vectors (i, j, k).

A: Understanding of cross product of three vectors are required to under this.

Q: Solve for the projection of the vector P = (3,4, 4) onto Q = (0, –2, –5). What is the exact measure…

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Q: Determine the value(s) of ? such that the vectors [5, ?, −8 ] and [3, ?, ?] are perpendicular.

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Q: Determine the angle between the given vectors. a = 3i – 2j, b = 2i + 2j (b) и %3D (1,—4, 2), v %3D…

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Q: Show that the pair of vectors is perpendicular. 3? + ? and ? − 3?

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Q: Find all unit vectors perpendicular to (3,1,4) that make equal angles with j and k.

A: Here we have to find the all unit vectors perpendicular to (3,1,4) that make equal angles with j and…

Q: Find the angle a between the vectors 5 2 and 1 a =

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Q: If a and b are unit vectors in R', show that ||a x b||? + (a · b)² = 1. %3D

A: On solving this we will get:-

Q: Prove that ? is parallel to vector ? using scaler product if ⃗̂̂ ?=−3?̂+4?̂−? ,?=−9?̂+12?̂−3?

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Q: ind the value of a so that vectors U = ai + 9j and V = 3i + 3j are

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Q: Find the volurne of the parallelepiped determined by the vectors a, b, and c. a = (6, 3, -2), b =…

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Q: For two nonzero vectors u and v, if u v=0 then u and v are parallel.

A: Given that u and v are two nonzero vectors   Dot product of two nonzero vectors u.v=|u|.|v|…

Q: Verify the parallelogram law for vectors u and v in R": |u + v|l? + ||u – v||° = 2||u||? + 2||v||?

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Q: If two vectors p and q are such that |p| = |q, show that p + q and p- q are (This proves that the…

A: Given p=qwe know that two vectors are perpendicular iff their dot product is zero

Q: If the volume of the parallelepiped determined by the vectors d, b, čeR' is a cubic units, which of…

A: Given, 2a→∧b→∧b→∧c→ Let, 2a→∧b→=u→ Now simplifying the given equation by apply the vector dot…

Q: Find the area of the parallelogram that has the vectors as adjacent sides.u = i − j + k, v = i + k

A: Given vectorsu = i-j+k , v = i+kArea of parallelogram is equal to u x v

Q: Find the angle θ between the vectors.u = (3, 1), v = (−2, 4)

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Q: (b) Check whether u = 6i – 2j+k and v = = -4i – 3j+9k are parallel or not. Then, find the area of…

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Q: Check whether u = 6i – 2j+k and v= -4i – 3j+9k are parallel or not. Then, find the area of triangle…

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Q: Find A:D,AAU ® let Ā = i+2j-K , B=-i+j+ K,itk> D=i+2j-K which vectors are: @ perpendiCular 6…

A: Please see the explanation below

Q: Find the measure of the angle between the two vectors. 24) u-(9,9) v-(-8,-5) V =

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Q: compute the dot products i ⋅ j, i ⋅ k, and j ⋅ k for the unit coordinate vectors. What do you…

A: Given: Compute the dot products i ⋅ j, i ⋅ k, and j ⋅ k for the unit coordinate vectors.

Q: Is the vector projection of B onto A, perpendicular to B? Is the angle between projAB and the vector…

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Q: Show that the indicated diagonal ofthe parallelogram determined by vectors u and v bisects the…

A: The given parallelogram is   Here, we are given |u|=|v| So, u+v is the required diagonal

Q: the angles between the vectors u = 2i - 2j+ k, v = 3i + 4k = O 84.2 O 40.2 O 48.2 80.3

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Q: Determine two unit vectors, vand wl, perpendicular to the plane passing through the z-axis and point…

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Q: and b are two unit vectors such that + 2b and 5 a. 4 b are perpendicular to each other, then the…

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Q: Suppose i and V are vectors such that ||ū|| = 2, |||| = 3, and the angle between u and ở is 60º. a)…

A: This question is related to vectors, we will use formula of dot product of vectors and basic…

Q: Find the value of a so that vectors U = ai + 12j and V = 31 + 9j are perpendicular. a =

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Q: 20. Show that the pair of vectors ' and ¯J is perpendicular.

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Q: The vertices of quadrilateral ABCD are given as A(0, 3, 5). B(3, – 1, 17), C(7, –3, 15), and D(4, 1,…

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Q: Assume that the vectors a, b, c, and d are defined as follows: a = (8,3) b = (6,5) c = (4, -1) d…

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Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.