Question

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

Answer #1

There is a mistake. The question should read “Find the
orthogonal projection of **v onto u**. Then write
**v** as the sum of two orthogonal vectors, one in
span{u} and one orthogonal to u.

We have proj_{v} (u) = [(v.u)/(u.u)]u =
[(6-10+3)/(4+25+1)]u = -(1/30)u = -(1/30) [2,-5,-1] =
[-1/15,1/6,1/30]. This vector, being a scalar multiple of u is in
span{u}.

Let x = [-1/15,1/6,1/30]. Then v-x = [3,2,-3]- [-1/15,1/6,1/30]= [ 46/15,11/6,-91/30] = y (say).

Then v = x+y, where x = [-1/15,1/6,1/30] is in span{u} and y = [ 46/15,11/6,-91/30] is orthogonal to u.

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

2.
a. Given u = (9,7) and v = (2,3), find the projection of u onto
v. (ordered pair)
b. Find the area of the parllelogram that has the given vectors
u = j and v = 2j + k as adjacent sides.

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.

Find the orthogonal projection of v=[−2,10,−16,−19] onto the
subspace W spanned by [-4,0,-2,1],[-4,-2,5,1],[3,-1,-3,4]

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).

Find the projection of u = −i
+ j + k onto v =
2i + j − 7k.

True or False
If A is the matrix of a projection onto a line L in R 2 and the
vector x in R 2 is not the zero vector, then the vector x − Ax is
perpendicular to the vector x.
If vectors u, v, x and y are vectors in R 7 such that u = 2v +
0x − 3y, then a basis for span(u, v, x, y) is {u, v, y}.

Find the orthogonal projection of v⃗ =⎢4,−11,−36,9⎤ onto the
subspace W spanned by ⎢0,0,−5,−2| , |−4,2,5,−5⎢ , ⎢−5,−5,0,5|

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.

1. Let ⃗u = −2[4,0,1]+[−1,3,−2] and ⃗v = 3[4,0,1]+5[−1,3,−2].
Let w⃗ = 3⃗u−⃗v. Express w⃗ as a linear combination of the vectors
[4, 0, 1] and [−1, 3, −2].
2. Let ⃗u and ⃗v be two vectors in Rn. Suppose that ||⃗u|| = 3,
||⃗u − ⃗v|| = 5, and that⃗u.⃗v = 1. What is ||⃗v||?.
3. Let ⃗u and ⃗v be two vectors in Rn. Suppose that ||⃗u|| = 5
and that ||⃗v|| = 2. Show that ||⃗u −...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 1 minute ago

asked 8 minutes ago

asked 18 minutes ago

asked 29 minutes ago

asked 30 minutes ago

asked 32 minutes ago

asked 35 minutes ago

asked 38 minutes ago

asked 38 minutes ago

asked 44 minutes ago

asked 44 minutes ago

asked 47 minutes ago