Question

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

Answer #1

Let u=[6,2 ], v=[3,3 ], and b=[4,1 ]. Find
(x⋅u+y⋅v-b)×2 u, where x,y are scalars.

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 ) − (...

Let U and V be vector spaces. Show that the Cartesian product U
× V = {(u, v) | u ∈ U, v ∈ V } is also a vector space.

Let vector u= 5i+3j+8k and vector v= i-j+2k
Find the component of v parallel to u and the component of v
perpendicular to u
find a unit vector perpendicular to both u and v

A rectangle with coordinate system with axes x,y,z is rotating
relative to an inertial frame with constant angular velocity w
about the z-axis. A particle of mass m moves under a force whose
potential is V(x,y,z). Set up the Lagrange equations of motion in
the coordinate system x,y,z. Show that their equations are the same
as those in a fixed coordinate system acted on by the force
-grad(V) and a force derivable from a velocity dependent potential
U. Find U.

It is not true that the equality u x (v x w) = (u x v) x
w for all vectors.
1. Find explicit vector for u, v and w where this equality does
not hold.
2. U, V and W are all nonzero vectors that satisfy the equality.
Show that at least one of the conditions below holds:
a) v is orthogonal to u and w.
b) w is a scalar multiple of u.
You can possibly use a...

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).

Let U and W be subspaces of a finite dimensional vector space V
such that V=U⊕W. For any x∈V write x=u+w where u∈U and w∈W. Let
R:U→U and S:W→W be linear transformations and define T:V→V by
Tx=Ru+Sw
.
Show that detT=detRdetS
.

One
electron and two protons in a x,y coordinate system. Find a point
with 0 V of potential. Fix a coordinate and solve for the other one
that gives 0 V.

Compute ∬S (x)(y^2)(z) dS where S is the part of the cone with
parameterization r(u,v)= <ucosv,usinv,u> , 0≤ u ≤ 1, 0≤ v ≤
pi/2 . Also state what the parameter space is.

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