Given ä = [1,3, -2] and b = [-2, 5,1). Determine: a) The Dot Product of the given vectors b) The Cross Product of å and b c) The Cross Product of B and å d) The magnitude of the sum of the given vectors e) The orthogonal vector to the given vectors f) The area of the parallelogram bound by the given vectors Show all your work. Clearly label answers for all cases (a, b, c..).

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

A:

Q: Find the area of the parallelogram that has the given vectors as adjacent sides. Use a computer…

A:

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 area of the triangle determined by the points P, Q, and R. Use * .the cross product of…

A:

Q: Find the area of the parallelogram that has the given vectors as adjacent sides. Use a computer…

A:

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

A:

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

A:

Q: ABC is an equilateral triangle of side 3 units. The points P, Q lie on BC, CA  respectively and are…

A:

Q: When two nonzero vectors w are called perpendicular or orthogonal?

A: Let two nonzero vectors are u and v Then the scalar product of the two vectors is given as,…

Q: If vector a= and vector b= are vectors a and b orthogonal, parallel, or neither? Provide

A:

Q: Consider vectors, 2 u = |2, v = w = |-3 (a) Do these vectors line in a plane ? Why? Why not? Find…

A:

Q: Suppose thatu and v are vectors in both of length and that the length of u-v is and the angle…

A: Let's find.

Q: The length of the cross product between the vectors j and -j is equal to the zero vector. Select…

A: Solutions

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 a parallelogram formed by the vectors PQ and PR, where P = (2, 5, 3), Q = (– 3, –…

A:

Q: . If u and v are nonzero vectors in Rº, why do the equa- tions u · (u x v) = 0 and v · (u x v) = 0…

A: The dot product of orthogonal vectors is equal to zero.

Q: Find the area of a parallelogram formed by the vectors PQ and PR, where P = (6, 6, 0), Q = (3, - 5,…

A: Let the parallelogram be PQRS, whose 3 vertices are P(6,6,0), Q(3,-5,6) & R(3,1,4). Then, these…

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

A:

Q: Determine which of the vectors is parallel to z and which is orthogonal to z if z = (3, 2) Show work…

A:

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: Show that the lines x = p + su and x = q + tv are skew lines. Find vector equations of a pair of…

A: Consider the matrix values given. The equation can be written as :

Q: u * v =( u1v1, UzV2 , Uz V3 , · Un Vn).

A:

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

A:

Q: 4 r 1 by -1 -2 -3 -4 -5 -4 -3 -2 -10 1 2 3 4 5

A: First we find p, q, r and s. Then solve it.

Q: Show that the lines L1 : x =1- t, y = 2t, z = 4+t and L2 : x = 1+ 3t, y = 3 – 6t, z = 5 – 3t are…

A:

Q: Let u and v be adjacent sides of a parallelogram. Use vectors to prove that the diagonals of the…

A: Vectors are quantities that have both magnitude and direction. We can add two vectors, and to the…

Q: 4. If two vectors P = 4i + 6j - 8k and F = 6i - 8j • 4k are added together describe the…

A: As we are solving only one problem at a time please submit the other question one by one as you…

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: We have, L1: x=4-t, y=2t, z=2+t and L2: x=1+4t, y=3-8t, z=5-4t

Q: The volume of the parallel pipe formed by three vectors A, B, and C is 50 cubic units. if A(-2,6,3),…

A:

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

A:

Q: Show that the lines L1 : x = 2 – t, y = 2t, z =1+t and L2 : x = 1+ 3t, y = 3 – 6t, z = 5 – 3t are…

A:

Q: given the vectors from R3 v, = (2 0 3), v2= ( -1 3 0), v3=(2 4 –1) %3D (-5 0 3) belongs to span{v,,…

A: Given vectors are v1→=2,0,3, v2→=-1,3,0, v3→=2,4,-1. To determine: u→=-5,0,3 belongs to span v1→,…

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

A: Area of the prallelogram using cross product

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

A:

Q: Show that the lines L1 : x = 3 – t, y = 2t, z = 3+t and L2 : x = 1+ 3t, y = 3 – 6t, z = 5 – 3t are…

A:

Q: Find the area of the parallelogram that has the vectors as adjacent sides.u = (2, −1, 0), v = (−1,…

A:

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

A: Let's find.

Q: Consider the vectors ā (2, 3, –1) and b = (4, –9, 6). Find the cross product of a and b

A:

Q: Find the area of the parallelogram that has the given vectors as adjacent sides. Use a computer…

A:

Q: Given that the set of vectors {v1, v2, v3}, where the vectors belong to R3 are linearly independent.…

A: We solve this by simple definition of linearly independent set

Q: Find the area of the parallelogram spanned by the following vectors. u = [4, –6, 5] v = [0, 6, 8]…

A:

Q: If L is a line in 2-space or 3-space that passes through the points A and B, then the distance from…

A:

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

A: To show that the lines L1: x=4-t, y=2t, z=5+t and L2: x=1+2t, y=3-4t, z=5-2t are parallel and to…

Q: value

A: If, P, Q & R are collinear, then,  The position vector of QP is, The position vector of P - the…

Q: 10. Calculate the area of the parallelogram formed by the vectors a =(3, 2,-2) and b = (2,-1,2).

A:

Q: Two vectars'a and -b are given as a =i+2j + kandb= 3i + j + 2k. The vector product ax.b is

A:

Q: If L is a line in 2-space or 3-space that passes through the points A and B, then the distance from…

A: Let us consider a line l joining by  two points A  and B. Consider a point P (not on line l ) then…

Q: Find the measure of the smallest nonnegative angle between the two vectors. Round approximate…

A:

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.