1. Compute the angle between the vectors u = [2, -1, 1] and and
v =...
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 u and v be vectors in 3-space with angle θ between them, 0 ≤
θ...
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...
Let u = <-2, 2> and v = <3,3>. Compute: a)
projv u b) Write u...
Let u = <-2, 2> and v = <3,3>. Compute: a)
projv u b) Write u = w1 +
w2, where w1 is parallel to v and
w2 is orthogonal to v.
1. Let ⃗u = −2[4,0,1]+[−1,3,−2] and ⃗v = 3[4,0,1]+5[−1,3,−2].
Let w⃗ = 3⃗u−⃗v. Express w⃗ as...
1. Let ⃗u = −2[4,0,1]+[−1,3,−2] and ⃗v = 3[4,0,1]+5[−1,3,−2].
Let w⃗ = 3⃗u−⃗v. Express w⃗ as a linear combination of the vectors
[4, 0, 1] and [−1, 3, −2].
2. Let ⃗u and ⃗v be two vectors in Rn. Suppose that ||⃗u|| = 3,
||⃗u − ⃗v|| = 5, and that⃗u.⃗v = 1. What is ||⃗v||?.
3. Let ⃗u and ⃗v be two vectors in Rn. Suppose that ||⃗u|| = 5
and that ||⃗v|| = 2. Show that ||⃗u −...
Let u = (−3, 2, 1, 0), v = (4, 7, −3, 2), w = (5,...
Let u = (−3, 2, 1, 0), v = (4, 7, −3, 2), w = (5, −2, 8, 1).
Find the vector x that satisfies 2u − ||v||v = 3(w − 2x).
Let g(u, v) = f(u 3 − v 3 , v3 − u 3 ). Prove...
Let g(u, v) = f(u 3 − v 3 , v3 − u 3 ). Prove that v^2 ∂g/∂u −
u^2 ∂g/∂v = 0, using the Chain Rule
Let U,V be i.i.d. random variables uniformly
distributed in [0,1]. Compute the following quantities:
E[|U−V|]=
P(U=V)=...
Let U,V be i.i.d. random variables uniformly
distributed in [0,1]. Compute the following quantities:
E[|U−V|]=
P(U=V)=
P(U≤V)=
Let V be the vector space of 2 × 2 matrices over R, let <A,
B>=...
Let V be the vector space of 2 × 2 matrices over R, let <A,
B>= tr(ABT ) be an inner product on V , and let U ⊆ V
be the subspace of symmetric 2 × 2 matrices. Compute the orthogonal
projection of the matrix A = (1 2
3 4)
on U, and compute the minimal distance between A and an element
of U.
Hint: Use the basis 1 0 0 0
0 0 0 1
0 1...
Let u=〈5,-1,6〉, v=〈0,1,2〉, and w=〈1,3,4〉. Find
(a)u×(v×w)
(b)(u×v)×w
(c)(u×v)×(v×w)
d)(v×w)×(u×v).
Let u=〈5,-1,6〉, v=〈0,1,2〉, and w=〈1,3,4〉. Find
(a)u×(v×w)
(b)(u×v)×w
(c)(u×v)×(v×w)
d)(v×w)×(u×v).