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Byju's Answer
Standard XII
Mathematics
Substitution Method to Remove Indeterminate Form
Find a →· b →...
Question
Find
a
→
·
b
→
when
(i)
a
→
=
i
^
-
2
j
^
+
k
^
and
b
→
=
4
i
^
-
4
j
^
+
7
k
^
(ii)
a
→
=
j
^
+
2
k
^
and
b
→
=
2
i
^
+
k
^
(iii)
a
→
=
j
^
-
k
^
and
b
→
=
2
i
^
+
3
j
^
-
2
k
^
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Solution
i
We have
a
→
=
i
∧
-
2
j
∧
+
k
∧
and
b
→
=
4
i
∧
-
j
∧
+7
k
∧
a
→
.
b
→
=
i
∧
-
2
j
∧
+
k
∧
.
4
i
∧
-
j
∧
+7
k
∧
=
1
4
+
-
2
-
4
+
1
7
=
4
+
8
+
7
=
19
ii
We have
a
→
=
j
^
+
2
k
^
=
0
i
∧
+
j
∧
+
2
k
∧
and
b
→
=2
i
∧
+
k
∧
=
2
i
∧
+0
j
∧
+
k
∧
a
→
.
b
→
=
0
i
∧
+
j
∧
+
2
k
∧
.
2
i
∧
+0
j
∧
+
k
∧
=
0
2
+
1
0
+
2
1
=
0
+
0
+
2
=
2
iii
We have
a
→
=
j
∧
-
k
∧
=
0
i
∧
+
j
∧
-
k
∧
and
b
→
=
2
i
∧
+
3
j
∧
-2
k
∧
a
→
.
b
→
=
0
i
∧
+
j
∧
-
k
∧
.
2
i
∧
+
3
j
∧
-2
k
∧
=
0
2
+
1
3
+
-
1
-
2
=
3
+
2
=
5
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Similar questions
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 angle between the vectors
a
→
and
b
→
, where
(i)
a
→
=
i
^
-
j
^
and
b
→
=
j
^
+
k
^
(ii)
a
→
=
3
i
^
-
2
j
^
-
6
k
^
and
b
→
=
4
i
^
-
j
^
+
8
k
^
(iii)
a
→
=
2
i
^
-
j
^
+
2
k
^
and
b
→
=
4
i
^
+
4
j
^
-
2
k
^
(iv)
a
→
=
2
i
^
-
3
j
^
+
k
^
and
b
→
=
i
^
+
j
^
-
2
k
^
(v)
a
→
=
i
^
+
2
j
^
-
k
^
,
b
→
=
i
^
-
j
^
+
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
^
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 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
^
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