Question

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=[k^{2}, 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] is a unit vector, find C. and v2 = [1/2, 3/2],

find v1 + v2 and graph it.

Answer #1

Find the angle theta between vectors u=(5,6) and v=(-8,7).
Find a unit vector orthogonal to v.

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

Suppose the vectors v1, v2, . . . , vp span a vector space V
.
(1) Show that for each i = 1, . . . , p, vi belongs to V ;
(2) Show that given any vector u ∈ V , v1, v2, . . . , vp, u also
span V

Do the vectors v1 = 1 2 3 ,
v2 = √ 3 √ 3 √ 3 ,
v3 √ 3 √ 5 √ 7 ,
v4 = 1 0 0 form a basis for R 3 ? Why or why not?
(b) Let V ⊂ R 4 be the subspace spanned by the vectors a1 and
a2, where a1 = ...

In
3 dimensions, draw vectors u, v, w, and x such that u+v+w=x. The
vectors u and x share an initium. You may pick the size of your
vectors. Make sure the math works.
Find the angle between vector x and vector u.

(a) Do the vectors v1 = 1 2 3 , v2 = √ 3 √ 3 √ 3 , v3=√ 3 √ 5 √
7, v4 = 1 0 0 form a basis for R 3 ? Why or why not?
(b) Let V ⊂ R 4 be the subspace spanned by the vectors a1 and
a2, where a1 = (1 0 −1 0) , a2 = 0 1 0 −1. Find a basis for the
orthogonal complement V ⊥...

1) If u and v are orthogonal unit vectors, under what condition
au+bv is orthogonal to cu+dv (where a, b, c, d are scalars)? What
are the lengths of those vectors (express them using a, b, c,
d)?
2) Given two vectors u and v that are not orthogonal, prove that
w=‖u‖2v−uuT v is orthogonal to u, where ‖x‖ is the L^2 norm of
x.

For the vectors Bold u equalsleft angle 3 comma 1 right angle
and Bold v equalsleft angle negative 1 comma negative 4 right
angle, express Bold u as the sum Bold u equalsBold pplusBold n,
where Bold p is parallel to Bold v and Bold n is orthogonal to Bold
v.

1) Find the angle θ between the vectors a=9i−j−4k and
b=2i+j−4k.
2) Find two vectors v1 and v2 whose sum is <-5,
2> where v1 is parallel to <-3 ,0> while v2 is
perpendicular to < -3,0>

2. Two vectors, ~v1 and ~v2. ~v1 has a length of 12 and is
oriented at an angle θ1 = 30o relative to the positive x−axis. ~v2
has a length of 6 and is oriented at an angle θ2 = 0o relative to
the positive x−axis (it is aligned with the positive x−axis)
a. What are the magnitude and direction (angle) of the sum of
the two vectors ( 1+2 ~v = ~v1 + ~v2)?
b. What are the magnitude...

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