If the volume of a parallelepiped whose coterminous edges are given by the vectors vec a=hat i+hat j+nhat k vec b=2i+4j-nhat k and vec c=hat i+hat j+3hat k(n>=0) is 158 cu.Units then
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Let the volume of a parallelopiped whose coterminous edges are given by
u
=
i
^
+
j
^
+λ
k
^
,
v
=
i
^
+
j
^
+3
k
^
and
w
=2
i
^
+
j
^
+
k
^
be 1cu. units. If θ be the angle between the edge
u
and
w
, then cosθ can be:
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Answer
Correct option is
D
6
3
7
Given,
u
=
i
^
+
j
^
+λ
k
^
,
v
=
i
^
+
j
^
+3
k
^
w
=2
i
^
+
j
^
+
k
^
Volume of parallelopied =u.(
v
×
w
)=[
u
v
w
]
=
∣
∣
∣
∣
∣
∣
∣
∣
1
1
2
1
1
1
λ
λ
1
∣
∣
∣
∣
∣
∣
∣
∣
=1(1−3)−1(1−6)+λ(1−2)−2+5+(−λ)3−λ
so,
Given volume =±1 taking (+) sign
3−λ=1
λ=2
Angle between
u
and
w
cosθ=
∣
u
∣.∣
w
∣
u
.
w
=
1+1+4
.
4+1+1
(i+j+2k).(2i+j+k)
=
5
.
5
2+1+2
=
5
5
=1
Taking (−) sign
3−λ=−1
λ=4
u
=
i
^
+
j
^
+4
k
^
w
=2i+j+
k
^
cosθ=
∣
u
∣∣
w
∣
u
.
w
=
1+1+16
4+1+1
(
i
^
+
j
^
+4
k
^
).(2i+j+k)
=
18
6
2+1+4
=
6
3
....Answer