show that the magnitude of resultant vector R is equal to √p2+q2+2pq cas thita
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Explanation:
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Class 11
>>Physics
>>Motion in a Plane
>>Position Vector and Displacement
>>The resultant of two vector...
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The resultant of two vectors
P
and
Q
is
R
. If the magnitude of
Q
is doubled, the new resultant vector becomes perpendicular to
P
. Then, the magnitude of
R
is equal to
Hard
Solution
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Correct option is
D
Q
Step 1: Initial resultant between two vectors [Ref. Fig. 1]
Resultant of
P
and
Q
,
∣
∣
∣
∣
R
∣
∣
∣
∣
2
=
∣
∣
∣
∣
P
∣
∣
∣
∣
2
+
∣
∣
∣
∣
Q
∣
∣
∣
∣
2
+2
∣
∣
∣
∣
P
∣
∣
∣
∣
∣
∣
∣
∣
Q
∣
∣
∣
∣
cosθ ....(1)
Step 2: Dot product between new resultant and
P
[Ref. Fig. 2]
When
Q
is doubled, new resultant,
R
1
=
P
+2
Q
becomes perpendicular to
P
∴
R
1
.
P
=0
⇒ (
P
+2
Q
).
P
=0
⇒ ∣
P
∣
2
+2∣
P
∣∣
Q
∣cosθ=0
⇒ cosθ=
2∣
Q
∣
−∣
P
∣
Step 3: Solving equation
Putting value of cosθ in equation (1)
∣
R
∣
2
=∣
P
∣
2
+∣
Q
∣
2
+2∣
P
∣∣
Q
∣[
2∣
Q
∣
−∣
P
∣
]
⇒ ∣
R
∣
2
=∣
Q
∣
2
⇒ ∣
R
∣=∣
Q