Question

Prove the simple, but extremely useful, result that the perpendicular distance from a point (u, v) to a line ax + by + c = 0 is given by |au + bv + c| if a 2 + b 2 = 1.

Answer #1

BY THIS WAY, PROVE EQUATION OF GIVEN PROBLEM.

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.

Let V be a three-dimensional vector space with ordered basis B =
{u, v, w}.
Suppose that T is a linear transformation from V to itself and
T(u) = u + v,
T(v) = u, T(w) =
v.
1. Find the matrix of T relative to the ordered basis B.
2. A typical element of V looks like
au + bv +
cw, where a, b and c
are scalars. Find T(au +
bv + cw). Now
that you know...

Let the linear transformation T: V--->W be such that T (u) =
u2 If a, b are Real. Find T (au + bv) ,
if u = (x, y) v = (z, w) and uv = (xz-yw, xw + yz)
Let the linear transformation T: V---> W be such that T (u)
= T (x, y) = (xy, 0) where u = (x, y), with 2, -3. Then, if u = (
1.0) and v = (0.1). Find the value...

Let u = (1,−3,3,9) and v = (2,1,0,−2). Find scalars a and b so
that au + bv = (−3,−5,3,10)

Prove the existence and uniqueness of a perpendicular drawn from
a point P to a line l.

please answer all of them
a. Suppose u and v are non-zero, parallel vectors. Which of the
following could not possibly be true?
a)
u • v = |u | |v|
b)
u + v = 0
c)
u × v = |u|2
d)
|u| + |v| = 2|u|
b. Given points A(3, -4, 2) and B(-12, 16, 12), point P, lying
between A and B such that AP= 3/5AB would have coordinates
a)
P(-27/5, 36/5, 42/5)
b)
P(-6, 8,...

On set R2 define ∼ by writing(a,b)∼(u,v)⇔ 2a−b =
2u−v. Prove that∼is an equivalence relation on R2
In the previous problem:
(1) Describe [(1,1)]∼. (That is formulate a statement P(x,y)
such that [(1,1)]∼ = {(x,y) ∈ R2 | P(x,y)}.)
(2) Describe [(a, b)]∼ for any given point (a, b).
(3) Plot sets [(1,1)]∼ and [(0,0)]∼ in R2.

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]...

1. Determine whether the lines are parallel, perpendicular or
neither. (x-1)/2 = (y+2)/5 = (z-3)/4 and (x-2)/4 = (y-1)/3 =
(z-2)/6
2. A) Find the line intersection of vector planes given by the
equations -2x+3y-z+4=0 and 3x-2y+z=-2
B) Given U = <2, -3, 4> and V= <-1, 3, -2> Find a. U
. V b. U x V

(a) Prove that if two linear transformations T,U : V --> W
have the same values on a basis for V, i.e., T(x) = U(x) for all x
belong to beta , then T = U. Conclude that every linear
transformation is uniquely determined by the images of basis
vectors.
(b) (7 points) Determine the linear transformation T : P1(R)
--> P2(R) given by T (1 + x) = 1+x^2, T(1- x) = x by finding the
image T(a+bx) of...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 22 minutes ago

asked 30 minutes ago

asked 51 minutes ago

asked 54 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 3 hours ago

asked 3 hours ago

asked 4 hours ago