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Byju's Answer
Standard XII
Mathematics
Archimedes's Method
Find the area...
Question
Find the area of the parallelogram determined by the vectors:
(i)
2
i
^
and
3
j
^
(ii)
2
i
^
+
j
^
+
3
k
^
and
i
^
-
j
^
(iii)
3
i
^
+
j
^
-
2
k
^
and
i
^
-
3
j
^
+
4
k
^
(iv)
i
^
-
3
j
^
+
k
^
and
i
^
+
j
^
+
k
^
.
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Solution
i
Let
:
a
=
2
i
^
+
0
j
^
+
0
k
^
b
→
=
0
i
^
+
3
j
^
+
0
k
^
∴
a
→
×
b
→
=
i
^
j
^
k
^
2
0
0
0
3
0
=
0
-
0
i
^
-
0
-
0
j
^
+
6
-
0
k
^
=
0
i
^
+
0
j
^
+
6
k
^
Area of the parallelogram=
a
→
×
b
→
=
0
+
0
+
6
2
=
6
sq. units
ii
Let
:
a
→
=
2
i
^
+
j
^
+
3
k
^
b
→
=
i
^
-
j
^
+
0
k
^
∴
a
→
×
b
→
=
i
^
j
^
k
^
2
1
3
1
-
1
0
=
0
+
3
i
^
-
0
-
3
j
^
+
-
2
-
1
k
^
=
3
i
^
+
3
j
^
-
3
k
^
Area of the parallelogram =
a
→
×
b
→
=
3
2
+
3
2
+
3
2
=
27
=
3
3
sq. units
iii
Let
:
a
→
=
3
i
^
+
j
^
-
2
k
^
b
→
=
1
i
^
-
3
j
^
+
4
k
^
a
→
×
b
→
=
i
^
j
^
k
^
3
1
-
2
1
-
3
4
=
i
^
4
-
6
-
j
^
12
+
2
+
k
^
-
9
-
1
=
-
2
i
^
-
14
j
^
-
10
k
^
Area of the parallelogram=
a
→
×
b
→
=
-
2
2
+
-
14
2
+
-
10
2
=
300
=
10
3
sq. units
iv
Let
:
a
→
=
i
^
-
3
j
^
+
k
^
b
→
=
i
^
+
j
^
+
k
^
a
→
×
b
→
=
i
^
j
^
k
^
1
-
3
1
1
1
1
=
-
3
-
1
i
^
-
1
-
1
j
^
+
1
+
3
k
^
=
-
4
i
^
+
0
j
^
+
4
k
^
Area of the parallelogram=
a
→
×
b
→
=
-
4
2
+
0
+
4
2
=
32
=
4
2
sq. units.
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0
Similar questions
Q.
Find the area of the parallelogram determined by the vectors:
(i)
2
i
^
and
3
j
^
(ii)
2
i
^
+
j
^
+
3
k
^
and
i
^
-
j
^
(iii)
3
i
^
+
j
^
-
2
k
^
and
i
^
-
3
j
^
+
4
k
^
(iv)
i
^
-
3
j
^
+
k
^
and
i
^
+
j
^
+
k
^
.
Q.
Find the area of the parallelogram whose diagonals are:
(i)
4
i
^
-
j
^
-
3
k
^
and
-
2
j
^
+
j
^
-
2
k
^
(ii)
2
i
^
+
k
^
and
i
^
+
j
^
+
k
^
(iii)
3
i
^
+
4
j
^
and
i
^
+
j
^
+
k
^
(iv)
2
i
^
+
3
j
^
+
6
k
^
and
3
i
^
-
6
j
^
+
2
k
^
Q.
Find
a
→
b
→
c
→
, when
(i)
a
→
=
2
i
^
-
3
j
^
,
b
→
=
i
^
+
j
^
-
k
^
and
c
→
=
3
i
^
-
k
^
(ii)
a
→
=
i
^
-
2
j
^
+
3
k
^
,
b
→
=
2
i
^
+
j
^
-
k
^
and
c
→
=
j
^
+
k
^
(iii)
a
→
=
2
i
^
+
3
j
^
+
k
^
,
b
→
=
i
^
-
2
j
^
+
k
^
and
c
→
=
-
3
i
^
+
j
^
+
2
k
^
Q.
Find the volume of the parallelopiped whose coterminous edges are represented by the vectors:
(i)
a
→
=
2
i
^
+
3
j
^
+
4
k
^
,
b
→
=
i
^
+
2
j
^
-
k
^
,
c
→
=
3
i
^
-
j
^
+
2
k
^
(ii)
a
→
=
2
i
^
-
3
j
^
+
4
k
^
,
b
→
=
i
^
+
2
j
^
-
k
^
,
c
→
=
3
i
^
-
j
^
-
2
k
^
(iii)
a
→
=
11
i
^
,
b
→
=
2
j
^
,
c
→
=
13
k
^
(iv)
a
→
=
i
^
+
j
^
+
k
^
,
b
→
=
i
^
-
j
^
+
k
^
,
c
→
=
i
^
+
2
j
^
-
k
^
Q.
Prove that the following vectors are coplanar:
(i)
2
i
^
-
j
^
+
k
^
,
i
^
-
3
j
^
-
5
k
^
and
3
i
^
-
4
j
^
-
4
k
^
(ii)
i
^
+
j
^
+
k
^
,
2
i
^
+
3
j
^
-
k
^
and
-
i
^
-
2
j
^
+
2
k
^
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