Q1. Given three vectors A = -a,+2a,+3a, B = 3a,+4a,+5a, and C = 2a, -2a,+7a,, compute : a) Angle between A and B b) Scalar projection of A on B Vector projection of B on A d) Area of the parallelogram whose sides are specified by A %3D and B

Q: Show that the lines L1 : x = 4 – t, y = 2t, z = 2+t and L2 : x = 1+ 4t, y = 3 – 8t, z = 5 – 4t are…

A:

Q: or what values of aj are the vectors [ai,4,3] and a) orthogonal? b) Parallel? c) Have 45 degree…

A: If 2 vectors a⇀,b⇀  are orthogonal, their dot product a⇀.b⇀ =0 Let u=a1,4,3 and v= 3,-2,12 So,…

Q: If the volume of the parallelepiped determined by the vectors A = i +ak, B = 8i-5j + ak and C = 2i…

A: Given vectors are A = i + αkB = 8i - 5j + αkC = 2i + 3j - 8k

Q: 1.) Find the area of the parallelogram with vectors as adjacent sides. и %3D (-2,9, —3) аnd v %3D…

A:

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: If a and b are two non-parallel vectors and the position vectors of P, Q and R are a + 2b,…

A: Given that P , Q & R lie on a straight line. It means vectors are coplanar . since if vectors…

Q: Find the area of the parallelogram determined by the vectors: u = (1,0,–3),v = (2,3,0)

A: By using formula  Area of parallelogram

Q: Find the volume of the parallelepiped with sides give by the three vectors a = 2i – 2j+ 3k, b = 5i+…

A: a→ = 2i^ -2j^ +3k^b→ = 5i^ +2j^ +6k^c→ =-3i^ +2j^ -2k^Volume of parallellpiped = a.(bx c¯

Q: Find the volume of the parallelogram in which the vectors (S, T, R) are adjacent sides whose values…

A: Given: The volume of the parallelogram of the vectors (S, T, R) that are adjacent sides can be…

Q: Find the area of the parallelogram that has the vectors u and v as adjacent sides, where ||u|| = 2,…

A: The area of the parallelogram is ∥u∥×∥v∥×sinθ=2×12×sinπ3=24×32=123   Here θ= the angle between u and…

Q: given two vectors v m which are perpendicular. under which condition is V+W perpendicular to V-W?

A: Since vectors v and w are perpendicular so, their dot product is 0

Q: Use the fact that the volume of a tetrahedron with adjacent edges given by the vectors u, v, and w…

A:

Q: If the projections of vector a on x,y and z −axes are 2,1,2 units respectively, find the angle at…

A:

Q: 1. Calculate the area of the parallelogram spanned by the vectors Á = and B = .

A:

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: Show that the lines L:a =4- t, y = 2t, z =1+t and L2:2 =1+ 21, y = 3 – 4t, z = 5 – 2t are parallel…

A: This is a problem of 3D analytical geometry and application of calculus, vector calculus.

Q: Find the area of the parallelogram that has the following vectors as adjacent sides. u = & v =

A:

Q: Find the volume of the parallelepiped determined by the vectors a, b, and c: a =, b =, c =.

A:

Q: If the volume of the parallelepiped determined by the vectors A = i + ak, B = 8i - 5j + ak and C =…

A:

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

A:

Q: Show that the points with position vectors 3i−5j, 4i+j, 6i+13j are collinear. If the point whose…

A:

Q: ABC is a triangle and P, Q are the midpoints of AB, AC respectively. If AB = 2x and AC = 2y, express…

A: Given P,Q are the mid-points of 2 sides AB,AC of the triangle ABC. Also, AB=2x, AC=2y.

Q: The area of a parallelopiped that is formed by two vectors A and B can be obtained mathematically…

A: It's area of parallelopiped is given by A×B.

Q: 2. The cross product of two vectors u and v is. to u and to v. Thus if both vectors u and v lie in a…

A: Given two vectors u and v

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

A:

Q: Let C and D be points with position vectors c and d respectively. If |c| = 5, \d| = 7 and c·d = 4,…

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: Find the area of a parallelogram formed by the vectors PQ and PR, where P = (- 3, 4, 3), Q = (6, 6,…

A: .

Q: The inner(dot) product of two vectors and with lengths a and b. is given by a-b = abcos(y+2 T) ,…

A: The statement is true. The given statement is true because , Given a→.b→=abcosγ+2π But, cos…

Q: The area of the parallelogram whose adjacent sides are the vectors u and v is a) Ju · v| c) Ju x v|…

A:

Q: A=\+2) -Kg B=-i+j+ K, Csitk, D= i+2j-K which vectors are: @ perpendicular parallel-

A:

Q: Given points P = (2,1,5), Q = (-1,3,4), R = (3,0,6), af Find the unit vector orthogonat…

A:

Q: Given that the vectors ở = (1, 2– k,3) and b = (7, 4, k+2) are perpendicular, what is the value of…

A: Given vectors are: a→=1,2-k,3 and b→=7,4,k+2

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

A:

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: After decompsing the Veclor A-2i-j+3k to Hhe Sum of tado vectors, the first is Parallel to the…

A: Every vector can be decomposed into two vectors. One vector is parallel to another given vector, and…

Q: Find the area of a parallelogram formed by the vectors PQ and PR, where P = (– 6, – 1, 3), Q = (- 1,…

A: The corresponding position vectors with respect to origin are,…

Q: Given two vectors a = (−2, 1, 3) and b = (3, −1, 2), does c = (−18, 7, 1) lie on the plane…

A:

Q: Suppose that u and v are vectors in R' both of length 2V2 v is also? v2. Thenllu + v|| and that the…

A:

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: Find the area of the parallelogram that has the vectors u = (5,-1, 4) and v = (2,-3, 1) as adjacent…

A:

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

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

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

A:

Q: Find the signed volume of the slanted box in R“ generated by the vectors 27 11 11 -2 3 , U2 = V3 =…

A: Form a matrix of the given vectors, A=v1v2v3v4=0-2-1-12-1-236-10-23030 The volume of the slanted box…

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

A:

Q: Find the signed volume of the slanted box in R" generated by the vectors 27 11 3 vi = 6. V3 = ,V4 =…

A: Given that, v1=0263,v2=-2-1-10,v3=-1-203,v4=-13-20

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.