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NDA Math Vector Algebra
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© examsnet.com
Question : 21
Total: 101
If
→
a
,
→
b
and
→
c
are coplanar, then what is
(
2
→
a
×
3
→
b
)
⋅
4
→
c
+
(
5
→
b
×
3
c
)
⋅
6
→
a
equal to?
[NDA-I 2021]
114
66
0
-66
Validate
Solution:
👈: Video Solution
CONCEPT:
Properties of Scalar Triple Product -
[
a
b
c
]
=
[
b
c
a
]
=
[
c
a
b
]
-
[
a
b
c
]
=
−
[
b
a
c
]
=
−
[
c
b
a
]
=
−
[
a
c
b
]
-
[
(
a
+
b
)
c
d
]
=
[
a
c
d
]
+
[
b
c
d
]
-
[
λ
a
,
b
c
]
=
λ
[
a
b
c
]
- Three non-zero vectors
→
a
,
→
b
and
→
c
are coplanar if and only if
[
a
b
c
]
=
0
CALCULATION:
Given:
→
a
,
→
b
and
→
c
are coplanar i.e
[
→
a
→
b
→
c
]
=
0
⇒
(
2
a
→
a
×
3
→
b
)
⋅
4
→
c
=
[
2
a
→
b
→
b
4
c
]
and
(
5
→
b
×
3
c
]
⋅
6
a
→
a
=
[
5
→
b
3
→
c
6
a
]
⇒
(
2
a
→
a
×
3
→
b
)
⋅
4
→
c
+
(
5
b
→
b
×
3
c
→
c
)
⋅
6
→
a
=
[
2
a
→
a
3
→
b
4
c
]
+
[
5
→
b
3
→
c
6
a
]
As we know that,
[
a
b
c
]
=
[
b
c
a
]
=
[
c
a
b
]
⇒
(
2
→
a
×
3
→
b
)
⋅
4
→
c
+
(
5
→
b
×
3
c
]
b
⋅
6
→
a
=
[
2
→
a
3
→
b
4
c
]
+
[
6
→
a
5
→
b
3
c
]
As we know that,
[
λ
a
,
b
c
]
=
λ
[
a
b
c
]
⇒
[
2
→
a
3
→
b
4
c
]
+
[
6
→
a
5
→
b
3
c
]
=
24
[
→
a
→
b
c
]
+
90
[
a
→
b
→
c
]
As we know that, vectors
→
a
,
→
b
and
→
c
are coplanar if and only if
[
a
b
c
]
=
0
⇒
[
2
→
a
3
→
b
4
c
]
+
[
6
a
→
a
5
→
b
3
c
]
=
0
⇒
(
2
→
a
×
3
→
b
)
⋅
4
→
c
+
(
5
→
b
×
3
c
]
⋅
6
→
a
=
0
Hence, correct option is 3 .
© examsnet.com
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