Question

For parts ( a ) − ( c ) , let u = 〈 2 , 4 , − 1 〉 and v = 〈 4 , − 2 , 1 〉 .

( a ) Find a unit vector which is orthogonal to both u and v .

( b ) Find the vector projection of u onto v .

( c ) Find the scalar projection of u onto v .

For parts ( a ) − ( c ) , let u = 〈 2 , 4 , − 1 〉 and v = 〈 4 , − 2 , 1 〉 .

( a ) Find a unit vector which is orthogonal to both u and v .

( b ) Find the vector projection of u onto v .

( c ) Find the scalar projection of u onto v .

Answer #1

Let u = ⟨1,3⟩ and v = ⟨4,1⟩.
(a) Find an exact expression and a numerical approximation for
the angle between u and v. (b) Find both the projection of u onto v
and the vector component of u orthogonal to v.
(c) Sketch u, v, and the two vectors you found in part
(b).

Consider the following.
u =
−6, −4, −7
, v =
3, 5, 2
(a) Find the projection of u onto
v.
(b) Find the vector component of u orthogonal to
v.

Given vector ? = 2? + 3?, ? = −5? + ? + ?. Find the
followings.
a) The projection of u onto v
b) A vector that is orthogonal to both u and v

U= [2,-5,-1] V=[3,2,-3] Find the orthogonal projection of u onto
v. Then write u as the sum of two orthogonal vectors, one in
span{U} and one orthogonal to U

Let u = [-2,1,3,1] and let v = [1,4,0,1]. a. Determine the
projector P1 that projects onto the subspace S1 spanned by the
vector u. What isthe rank of P1? b. Determine the projector that
projects onto the orthogonal complement of S1. c. Determine the
projector P2 that projects onto the subspace S2 spanned y the
vectors {u,v}. What is the rank of P2? d. Determine an orthogonal
projector that projects onto the orthogonal complement of S2. e.
Verify that...

Let V be the vector space of 2 × 2 matrices over R, let <A,
B>= tr(ABT ) be an inner product on V , and let U ⊆ V
be the subspace of symmetric 2 × 2 matrices. Compute the orthogonal
projection of the matrix A = (1 2
3 4)
on U, and compute the minimal distance between A and an element
of U.
Hint: Use the basis 1 0 0 0
0 0 0 1
0 1...

Find the orthogonal projection of u onto the
subspace of R4 spanned by the vectors
v1, v2 and
v3.
u = (3, 4, 2, 4) ;
v1 = (3, 2, 3, 0),
v2 = (-8, 3, 6, 3),
v3 = (6, 3, -8, 3)
Let (x, y, z, w) denote the
orthogonal projection of u onto the given
subspace. Then, the components of the target orthogonal projection
are

Let W be the subspace of R4 spanned by the vectors a = 3e1 − 4e2
and b = e2 + e3 + e4. Find the orthogonal projection of the vector
v = [2, 0, 1, 0] onto W. Then calculate the distance of the point v
from the subspace W.

let x=u+v. y=v find dS, the a vector, and ds^2 for the u,v
coordinate system and show that it is not an orthogonal system

1. Compute the angle between the vectors u = [2, -1, 1] and and
v = [1, -2 , -1]
2. Given that : 1. u=[1, -3] and v=[6, 2], are u and v
orthogonal?
3. if u=[1, -3] and v=[k2, k] are orthogonal vectors.
What is the
value(s) of k?
4. Find the distance between u=[root 3, 2, -2] and v=[0, 3,
-3]
5. Normalize the vector u=[root 2, -1, -3].
6. Given that: v1 = [1, - C/7]...

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