Question

-Let a and b be the vectors a = (1,−2,2) and b = (−3,4,0).

(a) Compute a · b.

(b) Find the angle between a and b. Leave it as an exact
answer.

(c) Compute a × b.

(d) Find the area of the parallelogram spanned by a and b.

(e) True or False: The product a · (b × a) is a vector.

Answer #1

1) Consider two vectors A=[20, 4, -6] and B=[8, -2, 6].
a) compute their dot product A.B
b) Compute the angle between the two vectors.
c)Find length and sign of component of A over B (mean Comp A
over B)and draw its diagram.
d) Compute Vector projection of B over A (means Proj B over A)
and draw corresponding diagram.
e) Compute Orthogonal projection of A onto B.

Let the vectors a and b be: a = <0,-3,4> and
b=<1,2,-2>. Find the following:
a) The vector c=2a+b and its length.
b) The cosine of the angle between the vector c and the y
axis
c) One unit vector that is orthogonal to both a and b.

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 =...

True or False
A.The dot product of two vectors is always less than the
product of the lengths of the vectors.
B. All linear combinations of the vectors u= (u1,u2) and
v= (v1,v2) form the parallelogram whose adjacent sides are u and
v.
C. If A and B are square matrices of the same size, then
the second column of AB can be obtained by multiplying the second
column of A to the matrix B.
D. The vector that starts...

Let W be the subspace of R4 spanned by the vectors a = 3e1 − 4e2
and b = e2 + e3 + e4. Find the orthogonal projection of the vector
v = [2, 0, 1, 0] onto W. Then calculate the distance of the point v
from the subspace W.

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]...

Create vectors x=(1,2,...,10) and y=(2,2^2,...,2^10). Then compute
the Euclidean distance and inner product between x and y,
respectively.
( Use R language and Do Not use "for")

1. Answer the following.
a. Find the area of a triangle that has sides of lengths 9, 10
and 13 inches.
b. True or False? If a, b, and θ are two sides and an included
angle of a parallelogram, the area of the parallelogram is
absinθ.
c. Find the smallest angle (in radians) of a triangle with sides
of length 3.6,5.5,3.6,5.5, and 4.54.5 cm.
d. Given △ABC with side a=7 cm, side c=7 cm, and angle B=0.5
radians, find...

Consider the following vectors
d1 : d1 = (23.0 i + 32.0 j) m
d2 : |d2| = 21.0 m and d2 makes an angle of 149° with the
positive x-axis.
Find the following quantities:
A. The magnitude of d1 and the angle θ between d1 and the
positive x-axis.
B. The resultant vector d = d1 + d2 in unit vector notation.
C. The scalar product between d1 and d2.
D. The vector product between d1 and d2.
E....

Cartesian Vectors (Write True or False)
1. Geometric vectors have both magnitude and direction but
Cartesian vectors have only magnitude.
2. Unit vectors have a magnitude of one but no direction
3. The product of a vector and scalar is a vector.
4. There is no such thing as a zero vector.
5. A Cartesian vector uses coordinates to describe it fully
6. .The dot product of two vectors is a vector.

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