For the following exercises, the vectors a, b, and c are given. Determine the vectors ( a ⋅ b ) c and ( a ⋅ c ) b . Express the vectors in component form. 127. a = 〈 2 , 0 , − 3 〉 , b = 〈 − 4 , − 7 , 1 〉 , c = 〈 1 , 1 , − 1 〉
For the following exercises, the vectors a, b, and c are given. Determine the vectors ( a ⋅ b ) c and ( a ⋅ c ) b . Express the vectors in component form. 127. a = 〈 2 , 0 , − 3 〉 , b = 〈 − 4 , − 7 , 1 〉 , c = 〈 1 , 1 , − 1 〉
For the following exercises, the vectors
a,
b,
and
c
are given. Determine the vectors
(
a
⋅
b
)
c
and
(
a
⋅
c
)
b
. Express the vectors in component form.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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Let n = 7, let p = 23 and let S be the set of least positive residues mod p of the first (p − 1)/2
multiple of n, i.e.
n mod p, 2n mod p, ...,
p-1
2
-n mod p.
Let T be the subset of S consisting of those residues which exceed p/2.
Find the set T, and hence compute the Legendre symbol (7|23).
23
32
how come?
The first 11 multiples of 7 reduced mod 23 are
7, 14, 21, 5, 12, 19, 3, 10, 17, 1, 8.
The set T is the subset of these residues exceeding
So T = {12, 14, 17, 19, 21}.
By Gauss' lemma (Apostol Theorem 9.6),
(7|23) = (−1)|T| = (−1)5 = −1.
Let n = 7, let p = 23 and let S be the set of least positive residues mod p of the first (p-1)/2
multiple of n, i.e.
n mod p, 2n mod p, ...,
2
p-1
-n mod p.
Let T be the subset of S consisting of those residues which exceed p/2.
Find the set T, and hence compute the Legendre symbol (7|23).
The first 11 multiples of 7 reduced mod 23 are
7, 14, 21, 5, 12, 19, 3, 10, 17, 1, 8.
23
The set T is the subset of these residues exceeding
2°
So T = {12, 14, 17, 19, 21}.
By Gauss' lemma (Apostol Theorem 9.6),
(7|23) = (−1)|T| = (−1)5 = −1.
how come?
Shading a Venn diagram with 3 sets: Unions, intersections, and...
The Venn diagram shows sets A, B, C, and the universal set U.
Shade (CUA)' n B on the Venn diagram.
U
Explanation
Check
A-
B
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田