let a and b be two given vectors such that |a| =2 and |b|= 1 and a×b=1 the angle between a and b is
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Answered by
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Step-by-step explanation:
Given ∣
a
∣=1,
∣
∣
∣
∣
b
∣
∣
∣
∣
=4
a
.
b
=2 and
c
=(2
a
×
b
)−3
b
a
.
b
=2⇒∣a∣∣b∣.cos(a,b)=2
1×4×cosθ=2
cosθ=
2
1
⇒θ=60
∘
(Angle b/w
a
,
b
)
a
×
b
=∣a∣∣b∣.sinθ=1×4×sin60=4×
2
2
=2
3
Given
c
=2
a
×
b
−3
b
squaring on both sides
∣
c
∣
2
=
∣
∣
∣
2
a
ˉ
×
b
ˉ
∣
∣
∣
2
+9
∣
∣
∣
∣
b
∣
∣
∣
∣
2
−2×(3
a
(2
a
×
b
))
∣
c
∣
2
=4
∣
∣
∣
∣
a
×
b
∣
∣
∣
∣
2
+9
∣
∣
∣
∣
b
∣
∣
∣
∣
2
−0
(∵
b
,2
a
×
b
are perpendicular to each other)
∣
c
∣
2
=4×(2
3
)
2
+9(4)
2
∣c∣
2
=4×12+9×16=48+144=192
∣c∣
2
=192⇒
∣c∣=8
3
c
=(2
a
×
b
)−3
b
dot product on both sides with
b
c
.
b
=(2
a
×
b
).
b
−3
b
.
b
c
.
b
=0−3∣b∣
2
(∵
b
12
a
×
b
and cos90=0)
∴∣c∣.∣b∣.cos(α)=−3∣b∣
cos(α)=
8
3
−3×4
=
2
−
3
−α=
6
5π
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