Homework problems 1. Verify the following identities for vectors v, w in R": (a) (av) (bw) = ab(v· w) (b) (av+bu) = a(u⋅ v) +b(u⋅ w) (c) |vw|² = |v|2 + |w² + 2 v⋅ w (d) v+w² – vw|² = 4v · w - (e) |√+w² + |v – w² = 2|v|² + 2 ||² 2. Verify the following "if and only if" statements for vectors u, w in R": (a) = 0 if and only if |+| = |√ – w| (b) = 0 if and only if |ʊ + w|² = |v|² + |w|² (c) vw=0 if and only if + cш| ≥ || for all values of the scalar CER (d) (v+w) (v– w) = 0 if and only if |σ| = ||| . 3. Assuming both and w are nonzero vectors, define their angle to be 0 = arccos ช.พี (a) Verify that the quantity inside the arccosine function lies in the domain [-1, +1] of that function. (b) Verify the following identity: | |√ – w|² = |v|² + ||² – 2 || || cos 0 (c) Why is this identity the same thing as the old "law of cosines" that you learned in trigonometry?
Homework problems 1. Verify the following identities for vectors v, w in R": (a) (av) (bw) = ab(v· w) (b) (av+bu) = a(u⋅ v) +b(u⋅ w) (c) |vw|² = |v|2 + |w² + 2 v⋅ w (d) v+w² – vw|² = 4v · w - (e) |√+w² + |v – w² = 2|v|² + 2 ||² 2. Verify the following "if and only if" statements for vectors u, w in R": (a) = 0 if and only if |+| = |√ – w| (b) = 0 if and only if |ʊ + w|² = |v|² + |w|² (c) vw=0 if and only if + cш| ≥ || for all values of the scalar CER (d) (v+w) (v– w) = 0 if and only if |σ| = ||| . 3. Assuming both and w are nonzero vectors, define their angle to be 0 = arccos ช.พี (a) Verify that the quantity inside the arccosine function lies in the domain [-1, +1] of that function. (b) Verify the following identity: | |√ – w|² = |v|² + ||² – 2 || || cos 0 (c) Why is this identity the same thing as the old "law of cosines" that you learned in trigonometry?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Homework problems
1. Verify the following identities for vectors v, w in R":
(a) (av) (bw) = ab(v· w)
(b) (av+bu) = a(u⋅ v) +b(u⋅ w)
(c) |vw|² = |v|2 + |w² + 2 v⋅ w
(d) v+w² – vw|² = 4v · w
-
(e) |√+w² + |v – w² = 2|v|² + 2 ||²
2. Verify the following "if and only if" statements for vectors u, w in R":
(a)
= 0 if and only if |+| = |√ – w|
(b)
= 0 if and only if |ʊ + w|² = |v|² + |w|²
(c) vw=0 if and only if + cш| ≥ || for all values of the scalar
CER
(d) (v+w) (v– w) = 0 if and only if |σ| = |||
.
3. Assuming both and w are nonzero vectors, define their angle to be
0 = arccos
ช.พี
(a) Verify that the quantity inside the arccosine function lies in the
domain [-1, +1] of that function.
(b) Verify the following identity:
| |√ – w|² = |v|² + ||² – 2 || || cos 0
(c) Why is this identity the same thing as the old "law of cosines"
that you learned in trigonometry?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe1d06251-9cb7-4f37-8e8f-dbbf0e17b22e%2F4366624d-ed7c-426b-9360-0af51c3045bd%2Fduqucyq_processed.png&w=3840&q=75)
Transcribed Image Text:Homework problems
1. Verify the following identities for vectors v, w in R":
(a) (av) (bw) = ab(v· w)
(b) (av+bu) = a(u⋅ v) +b(u⋅ w)
(c) |vw|² = |v|2 + |w² + 2 v⋅ w
(d) v+w² – vw|² = 4v · w
-
(e) |√+w² + |v – w² = 2|v|² + 2 ||²
2. Verify the following "if and only if" statements for vectors u, w in R":
(a)
= 0 if and only if |+| = |√ – w|
(b)
= 0 if and only if |ʊ + w|² = |v|² + |w|²
(c) vw=0 if and only if + cш| ≥ || for all values of the scalar
CER
(d) (v+w) (v– w) = 0 if and only if |σ| = |||
.
3. Assuming both and w are nonzero vectors, define their angle to be
0 = arccos
ช.พี
(a) Verify that the quantity inside the arccosine function lies in the
domain [-1, +1] of that function.
(b) Verify the following identity:
| |√ – w|² = |v|² + ||² – 2 || || cos 0
(c) Why is this identity the same thing as the old "law of cosines"
that you learned in trigonometry?
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