Question

5. a) Suppose that the area of the parallelogram spanned by the vectors ~u and ~v is 10. What is the area of the parallogram spanned by the vectors 2~u + 3~v and −3~u + 4~v ?

(b) Given (~u × ~v) · ~w = 10. What is ((~u + ~v) × (~v + ~w)) · ( ~w + ~u)? [4]

6. Find an equation of the plane that is perpendicular to the plane x + 2y + 4 = 0, contains the origin, and whose normal makes an angle of 30◦ with the z − axis.

Please answer both questions with detailed solution.

Answer #1

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.

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)

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

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,...

5.
Perform the following operations on the vectors u=(-4, 0, 2),
v=(-1, -5, -4), and w=(0, 3, 4)
u*w=
(u*v)u=
((w*w)u)*u=
u*v+v*w=

Let u and v be vectors in 3-space with angle θ between them, 0 ≤
θ ≤ π. Which of the following is the only correct statement?
(a) u × v is parallel to v, and |u × v| = |u||v| cos θ.
(b) u × v is perpendicular to u, and |u × v| = |u||v| cos θ.
(c) u × v is parallel to v, and |u × v| = |u||v|sin θ.
(d) u × v is perpendicular...

(a) Find the volume of the parallelepiped determined by the
vectors a =< 2, −1, 3 >, b =< −3, 0, 1 >, c =< 2, 4,
1 >.
(b) Find an equation of the plane that passes through the point
(2, 4, −3) and is perpendicular to the planes 3x + 2y − z = 1 and x
− 2y + 3z = 4.

Let u, v, and w be vectors in Rn. Determine which of the
following statements are always true. (i) If ||u|| = 4, ||v|| = 5,
and ?||u + v|| = 8, then u?·?v = 4. (ii) If ||u|| = 2 and ||v|| =
3, ?then |u?·?v| ? 5. (iii) The expression (v?·?w)u is both
meaningful and defined. (A) (ii) and (iii) only (B) (ii) only (C)
none of them (D) all of them (E) (i) only (F) (i) and...

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]...

Let
R4
have the inner product
<u, v> =
u1v1 +
2u2v2 +
3u3v3 +
4u4v4
(a)
Let w = (0, 6,
4, 1). Find ||w||.
(b)
Let W be the
subspace spanned by the vectors
u1 = (0, 0, 2,
1), and u2 = (3, 0, −2,
1).
Use the Gram-Schmidt process to transform the basis
{u1,
u2} into an
orthonormal basis {v1,
v2}. Enter the
components of the vector v2 into the
answer box below, separated with commas.

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