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Concept explainers
Writing
P=(1,2),Q=(4,1),R=(5,4)
(a)
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To calculate: For given P=(1,2),Q=(4,1),R=(5,4), u=→PQ,v=→PR Find acomponent form of u and v.
Answer to Problem 1RE
Solution:
For given P=(1,2),Q=(4,1),R=(5,4), u=→PQ,v=→PR
Find a component form of u and vis 〈3,−1〉 and 〈4,2〉 respectively.
Explanation of Solution
Given:
P=(1,2),Q=(4,1),R=(5,4), u=→PQ,v=→PR
Formula used:
If P(p1,p2) and Q(q1,q2) are the initial and terminal points of a directed line segment, then the component form of the vector u represented by →PQ is
〈u1,u2〉=〈q1−p1,q2−p2〉
Calculation:
If
u2=q2−p2=1−2=−1
And
u =→PQ and P=(1,2),Q=(4,1), then:
u1=q1−p1=4−1=3
The component form of u is 〈3,−1〉.
If v =→PR and P=(1,2),R=(5,4), then:
v2=r2−p2=4−2=2
And
v1=r1−p1=5−1=4
The component form of v is 〈4,2〉.
(b)
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To calculate: For given P=(1,2),Q=(4,1),R=(5,4), u=→PQ,v=→PR, find u and vas the linear combination of the standard unit vectors i and j.
Answer to Problem 1RE
Solution: For given P=(1,2),Q=(4,1),R=(5,4), u=→PQ,v=→PR, u and vas the linear combination of the standard unit vectors i and j are the vectors u=3i−j and v=4i+2j
Explanation of Solution
Given:
P=(1,2),Q=(4,1),R=(5,4), u=→PQ,v=→PR
Formula used:
If 〈v1,v2〉 is the component form of v, then the vector v=v1i+v2j is called a linear combination of i and j.
Calculation:
The vectors u=3i−j and v=4i+2j are the linear combinations of i and j.
As according to the calculation of part (a), a component form of u and vis 〈3,−1〉 and 〈4,2〉.
(c)
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To calculate: Forgiven P=(1,2),Q=(4,1),R=(5,4), u=→PQ,v=→PR. Find magnitudes of u and v.
Answer to Problem 1RE
Solution:
Forgiven P=(1,2),Q=(4,1),R=(5,4), u=→PQ,v=→PR
Magnitudes of u and vu is ‖u‖=√10 and magnitude of v is ‖v‖=2√5.
Explanation of Solution
Given:
P=(1,2),Q=(4,1),R=(5,4), u=→PQ,v=→PR
Formula used:
According to the Distance Formula, the length (or magnitude) of vectoru is:
‖u‖=√(q1−p1)2+(q2−p2)2=√u12+u22
Calculation:
As per part (a),
v=〈4,2〉. The length of vector v is:
‖v‖=√v12+v22=√42+22=√16+4=√20
u=〈3,−1〉. The length of vector u is:
‖u‖=√u12+u22=√32+(−1)2=√9+1=√10
and
Therefore, the magnitude of u is ‖u‖=√10 and magnitude of v is ‖v‖=2√5.
(d)
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To calculate: For given P=(1,2),Q=(4,1),R=(5,4), u=→PQ,v=→PR. Find the value of -3u+v.
Answer to Problem 1RE
Solution:
For given P=(1,2),Q=(4,1),R=(5,4), u=→PQ,v=→PR. The value of -3u+v is 〈−5,5〉.
Explanation of Solution
Given:
P=(1,2),Q=(4,1),R=(5,4), u=→PQ,v=→PR
Formula used:
The vector sum of u and v is the vector:
u+v=〈u1+v1,u2+v2〉
The scalar multiple of c and u is the vector:
cu=〈cu1,cu2〉
Calculation:
Value of -3u+v using scalar multiplication and vector sum formulae:
As according to the calculation of part (a), a component form of u and v is 〈3,−1〉 and 〈4,2〉.
-3u+v=−3〈3,−1〉+〈4,2〉=〈−9,3〉+〈4,2〉=〈−5,5〉
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Chapter 11 Solutions
Multivariable Calculus (looseleaf)
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