Cauchy–Schwarz Inequality The definition u · v = | u | | v | cos θ implies that | u · v | ≤ | u | | v | ( because | cos θ | ≤ 1). This inequality, known as the Cauchy–Schwarz Inequality, holds in any number of dimensions and has many consequences. 86. Geometric-arithmetic mean Use the vectors u = 〈 a , b 〉 and v = 〈 b , a 〉 to show that a b ≤ ( a + b ) / 2 , where a ≥ 0 and b ≥ 0.
Cauchy–Schwarz Inequality The definition u · v = | u | | v | cos θ implies that | u · v | ≤ | u | | v | ( because | cos θ | ≤ 1). This inequality, known as the Cauchy–Schwarz Inequality, holds in any number of dimensions and has many consequences. 86. Geometric-arithmetic mean Use the vectors u = 〈 a , b 〉 and v = 〈 b , a 〉 to show that a b ≤ ( a + b ) / 2 , where a ≥ 0 and b ≥ 0.
Solution Summary: The author explains that the Cauchy-Schwarz inequality is given by left|ucdot vright|le
Cauchy–Schwarz InequalityThe definitionu · v = |u| |v| cos θ implies that |u · v| ≤ | u| |v| (because | cos θ| ≤ 1). This inequality, known as the Cauchy–Schwarz Inequality, holds in any number of dimensions and has many consequences.
86. Geometric-arithmetic mean Use the vectors
u
=
〈
a
,
b
〉
and
v
=
〈
b
,
a
〉
to show that
a
b
≤
(
a
+
b
)
/
2
, where a ≥ 0 and b ≥ 0.
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.
University Calculus: Early Transcendentals (3rd Edition)
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.