Question

Find two vectors v¯¯¯1 and v¯¯¯2 whose sum is 〈5,1〉, where v¯¯¯1 is parallel to 〈5,−1〉 while v¯¯¯2 is perpendicular to 〈5,−1〉 . v¯¯¯1 = and v¯¯¯2 =

Answer #1

Find two vectors v¯1 and v¯2 whose sum is 〈2,3〉, where v¯1 is
parallel to 〈−1,3〉 while v¯2 is perpendicular to 〈−1,3〉.
v¯1 = and
v¯2 =

Find two vectors v1 and v2 whose sum is 〈1,3〉, where v1 is
parallel to 〈1,2〉 while v2 is perpendicular to 〈1,2〉.
v1 =
and
v2 =

1) Find the angle θ between the vectors a=9i−j−4k and
b=2i+j−4k.
2) Find two vectors v1 and v2 whose sum is <-5,
2> where v1 is parallel to <-3 ,0> while v2 is
perpendicular to < -3,0>

Find two non-parallel vectors v1 and v2 in R2 such that
||v1||2=||v2||2=1 and whose angle with [3,4] is 42 degrees. *Note:
[3,4] is a 2 by 1 matrix with 3 on the top and 4 on the bottom.

Decompose the vector w into a sum of its vectors in its parallel
and perpendicular components. The vector w is described as 3 in the
x direction and -4 in the y direction. The parallel component of
the x vector should be in the direction of the vector u and the
perpendicular component of vector w is perpendicular to vector u.
The vector u is described as 1 in the x direction and 2 in the y
direction.

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 =...

(a) Find the unit vectors that are parallel to the tangent line
to the curve
y = 2 sin x
at the point (π/6, 1). (Enter your answer as a
comma-separated list of vectors.)
(b) Find the unit vectors that are perpendicular to the tangent
line.

1) Find two positive numbers whose sum is 31 and product is
maximum.
2) Find two positive whose product is 192 and the sum is
minimum.

2. Given ⃗v = 〈3,4〉 and w⃗ = 〈−3,−5〉 find
(a) comp⃗vw⃗
(b) proj⃗vw⃗
(c) The angle 0 ≤ θ ≤ π (in radians) between ⃗v and w⃗.
1. Let d--> =2i−4j+k. Write⃗a=3i+2j−6k as the sum
of two vectors⃗v,w⃗, where⃗v is perpendicular
to d--> and w⃗ is parallel to
d-->.

U= [2,-5,-1] V=[3,2,-3] Find the orthogonal projection of u onto
v. Then write u as the sum of two orthogonal vectors, one in
span{U} and one orthogonal to U

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