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Mathematics
Mathematics
Problem
1. (i) Find the area of the quadrilateral whose diagonals are given by
3
i
^
+
j
^
−
2
k
^
,
i
^
−
3
j
^
+
4
k
^
3\hat{i} + \hat{j} - 2\hat{k}, \hat{i} - 3\hat{j} + 4\hat{k}
3
i
^
+
j
^
−
2
k
^
,
i
^
−
3
j
^
+
4
k
^
2. (ii)
A
1
,
A
2
,
…
,
A
n
A_1, A_2, \dots, A_n
A
1
,
A
2
,
…
,
A
n
are the vertices of a regular plane polygon with
n
n
n
sides.
O
O
O
is the centre. Show that
∑
i
=
1
n
−
1
(
O
A
i
×
O
A
i
+
1
)
=
(
1
−
n
)
(
O
A
2
×
O
A
1
)
\sum_{i=1}^{n-1} (\mathbf{OA}_i \times \mathbf{OA}_{i+1}) = (1-n)(\mathbf{OA}_2 \times \mathbf{OA}_1)
i
=
1
∑
n
−
1
(
OA
i
×
OA
i
+
1
)
=
(
1
−
n
)
(
OA
2
×
OA
1
)
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