Question

The three following coordinate vectors are given in unitary
coordinates (in [m]):

a = (5; 0; 0)

b = (-1; 4; 1)

c = (0; 1; 3)

a) Determine the new coordinate system, giving |a|, |b|, |c|, alpha, beta and gamma. Use vector operations to obtain the values.

b) Determine the metric matrix for this coordinate system, and the volume of the parallelepiped spanned by a, b, c. For the volume calculation, use the determinant of the metric matrix.

c) Calculate the distance between two end points of the vectors given in relative coordinates (u; v; w) (1/2, 1/2, 0) and (1/2, 1/4, 1/4), and determine the angle between them! Use the metric matrix for this calculation, and compare the results to the method using the unitary coordinate system.

d) Determine the associated reciprocal vectors a*, b*, c* and the reciprocal angles alpha*, beta*, gamma*. Use the vector relationships to calculate the vectors in the reciprocal unitary system. Determine the reciprocal coordinate system by inversion of the metric matrix of the direct lattice.

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Answer #1

Exercise 6. Consider the following vectors in R3 . v1 = (1, −1,
0) v2 = (3, 2, −1) v3 = (3, 5, −2 ) (a) Verify
that the general vector u = (x, y, z) can be written as a linear
combination of v1, v2, and v3. (Hint : The coefficients will be
expressed as functions of the entries x, y and z of u.) Note : This
shows that Span{v1, v2, v3} = R3 . (b) Can R3 be...

Problem 5
A surveying team records the following displacement vectors on
three days (24 hour days) of walking, each one in succession (1,
then 2, then 3) (Give 2 significant figures):
d1 = (7 i + 8 j)km, d2 = (6 i + 2 j)km, and d3 =
(2 i + 9 j)km.
(a): Sketch these vectors on a Cartesian plane. Hint: The
vectors should all be connected
head-to-tail.
(b):Write the total displacement vector d(total) for the
three days in...

Problem 5
A surveying team records the following displacement vectors on
three days (24 hour days) of walking, each one in succession (1,
then 2, then 3) (Give 2 significant figures):
d1 = (7 i + 8 j)km, d2 = (6 i + 2 j)km, and d3 =
(2 i + 9 j)km.
(a): Sketch these vectors on a Cartesian plane. Hint: The
vectors should all be connected
head-to-tail.
(b):Write the total displacement vector d(total) for the
three days in...

Here are some vectors in R 4 : u1 = [1 3 −1 1] u2 = [1 4 −1 1]
u3 = [1 0 −1 1] u4 = [2 −1 −2 2] u5 = [1 4 0 1]
(a) Explain why these vectors cannot possibly be
independent.
(b) Form a matrix A whose columns are the ui’s and compute the
rref(A).
(c) Solve the homogeneous system Ax = 0 in parametric form and
then in vector form. (Be sure the...

Three point charges ?1 = 5.00 ??, ?2 = −4.00 ??, and ?3 = 7.00
?? lie in an x-y coordinate system with coordinates ? = (1, 1), ? =
(2, 2), and ? = (4, 1) where the coordinate positions have units of
meters. Determine (a) the net force on charge ?3 in unit vector
notation, (b) the electric potential at the origin, and (c) the
work necessary to assemble this charge distribution. Give numerical
results with appropriate units...

1.
a) Find a value of x other than 0 such that the vectors <-3x,
2x> and <4, x> are perpendicular
b) find the domain of the vector function r (t) = <sin (t),
ln (t), 1 / (t-1)>
c) Determine if the sequence converges or diverges, if it
converges determines its limit
ln (2 + e ^ n) / 2020n
d) Find the point (a, b, c) where the line x = 1-t, y = t, z = 1...

Answer all of the questions true or false:
1.
a) If one row in an echelon form for an augmented matrix is [0 0 5
0 0]
b) A vector b is a linear combination of the columns of a matrix A
if and only if the
equation Ax=b has at least one solution.
c) The solution set of b is the set of all vectors of the form u =
+ p + vh
where vh is any solution...

Each of the following vectors is given in terms of its
x and y components. Find the magnitude of each
vector and the angle it makes with respect to the +x
axis.
a- AxAx = -1, AyAy = -6. Find the magnitude of this vector.
(Express your answer to two significant figures.)
b- AxAx = -1, AyAy = -6. Find the angle this vector
makes with respect to the +x axis. Use value from -180∘∘
to +180 ∘∘. (Express your...

1. A plane passes through A(1, 2, 3), B(1, -1, 0) and
C(2, -3, -4). Determine vector and parametric equations of
the plane. You must show and explain all steps for full marks. Use
AB and AC as your direction vectors and point A as your starting
(x,y,z) value.
2. Determine if the point (4,-2,0) lies in the plane with vector
equation (x, y, z) = (2, 0, -1) + s(4, -2, 1) + t(-3, -1,
2).

1. Find the area of the parallelogram that has the given vectors
as adjacent sides. Use a computer algebra system or a graphing
utility to verify your result.
u
=
3, 2, −1
v
=
1, 2, 3
3. Find the area of the triangle with the given vertices.
Hint:
1
2
||u ✕ v||
is the area of the triangle having u and
v as adjacent
sides.
A(4, −5, 6), B(0, 1, 2), C(−1, 2, 0)

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