Write down the 2 by 2 matrix that a) project every vector onto x¡ axis. b) rotate a vector clockwise for a given angle, 0.

Q: Find the unit vector in the direction of v and verify that it has length 1.

A: If vector u is given, then the unit vector in direction of vector u is given as,

Q: Given the vectors ū = (-2, 1, 1) and V = (3, –1, 1), calculate the vector projection of ū onto V

A:

Q: Use paper and pencil to identify the redundant vectors. 4 3. 0. 0.

A: Definition : Redundant vector in a set is a vector which can be written as linear combination of…

Q: Show that, û . ûm = 0, Use only the fact that û,is a unit vector. Interpret this result.

A: Here we need to prove that dot product of a unit vector and its derivative is zero. Here we use the…

Q: Given the vectors u = (2, -2, 3) and v = (-3, 2, – 1), calculate the vector projection of u onto v…

A:

Q: snip1

A: orthogonal to u means the dot product of the vectors should be zeroso v.u=0xy.31=03x+y=0from here we…

Q: Show that Il all = 11b|l Are the vectors ã' and b equal?

A: ||a|| = ||b|| a and b are not equal.

Q: Let j = | and ü .Write j as the sum of two orthogonal vectors, it -2. %3D an {u} and ¤2 orthogonal…

A: Given:- →y=-69  →u=5-2

Q: Determine the resulting vector when á = (-2,4) is rotated 60° clockwise and increased in size %3D by…

A:

Q: Find a unit vector in the direction of u = (3, 4, 5).

A: Given vector, u=3, 4, 5

Q: Find a unit vector that is orthogonal to both u and v. u = (-8, -6, 4) v = (17, -18, –1)

A: the given vectors are: u=-8,-6,4v=17,-18,-1 we have to find a unit vector that is orthogonal to both…

Q: Give vectors a, b, c, d such that a=5c + 3d and b= 2c - d express 2a - 3b in terms of c and d

A: Solution as given below:

Q: Find the vector z, given that u = , v = , and w = . z = 8u - 3V - -W 2 1. Z =

A: We have been given the vectors  u = <1, 2, 3> v = <2, 2, -1> w = <4, 0, -4>

Q: What is the definition of the triple scalar product for vectors u, v, and w in ℝ3?

A: To Define: Scalar Triple Product for vectors u, v and w in ℝ3.

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

A:

Q: Express the vector in the form (a, b). 1(i + j) ji + jl

A:

Q: Use paper and pencil to identify the redundant vectors. 2 1 3. 4. 3. 4.

A: We have to identify redundant vectors.

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

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Q: Write w as the sum of a vector in Span{v} and a vector orthogonal to

A:

Q: Find the vector of length v7 in the direction of AB + AC

A:

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: Find the following vector. The vector in the direction of (24,10) with length 2

A:

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

A:

Q: Find the unit vector u in the di rection of the vector v=2(2, 1, 2).

A:

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: Describe all vectors v that are orthogonal to u

A:

Q: Find a unit vector u in the direction opposite of (-8,-4,-3).

A:

Q: Determine whether the vectors AB and PQ are equal to each other.

A: Solution of the problem as follows

Q: Find the scalar and vector projections of b onto a, where a = (1, 2) and b = (-4, 1). 1. comp, b 2.…

A:

Q: For each pair of vectors, calculate projvu and the vector component of u orthogonal to v. (a) u =…

A: Vector components

Q: Le and ü .Write j as the sum of two orthogonal vectors, a, in Span {u} and i2 orthogonal to ū.

A:

Q: Find a unit vector in the direction of u = (5, 3, 2).

A: Given vector is u=(5,3,2) Find a unit vector

Q: Consider two vectors ~a and ~b such that ~a x^b = ~0. What can be said about vectors ~a and b?…

A: We have to give the reasons

Q: Given nonzero vectors u, v, and w, use dot product and crossproduct notation, as appropriate, to…

A: Let the two vectors be u=u1i+u2j+u3kv=v1i+v2j+v3k

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: B. Find a vector has length 10 in the direction of vector V=.

A:

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 if the vectors u=, *v= and w= lie in the same plane or not

A:

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: . CALCULATE the CROSS product for each of the following pairs of vectors, AND VERIFY your answer by…

A:

Q: Repeat the directions of Exercise 3 with the same vector w, but with u, = uz = -2 u = u u = |-3 9-…

A: Please find the answer in next step

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: Given nonzero vectors u, v, and w, use dot product and crossproduct notation, as appropriate, to…

A: It is known that the unit vector along the direction v is given by vv. Since it is an unit vector,…

Q: What is the vector j?

A: We have to tell what is vector j   it is also written as j→

Q: Find a vector a that has the same direction as (-10,7, 10) bưut has length 4.

A: In the given problem we have to find a vector a that has the same direction as -10, 7, 10 but has…

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 unit vector that is orthogonal to both u and v. u = (-8, -6, 4) (10, -12, -2) %3D V =

A: Find the unit vector that is orthogonal to both u and v

Q: Write y as the sum of two orthogonal vectors, x1 in Span{u} and x2 orthogonal to u

A:

Q: Find two vectors v and w such that the projection of v onto w is v.

A: Given: There are two vectors v and w such that the projection of v onto w is v. We have to find that…

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.