Determinants and areas
�
�
� 1 2
�
�
�.
1. a) Compute
�
3 4
�
�
�
� 1 −2 �
�
� .
b) Compute
�
−3
4 �
�
�
� 3 4
�
�
� .
c) Compute
�
1 2 �
�
�
� 1 2 �
�
�
Answer: a)
�
= 1 · 4 − 2 · 3 = −2.
3 4
�
�
�
� 1 −2 �
�
� = 1 · 4 − (−2) · (−3) = −2.
b)
�
−3
4 �
�
�
� 3 4
�
� = 3 · 2 − 4 · 1 = 2.
c)
��
1 2
�
2. Find the area of the quadrilateral shown.
y
(4, 3)
(1, 2)
x
(3, −1)
Answer:
y
B
A
�1, 2�
�4, 3�
x
O
�3, −1�
C
We break the quadrilateral into two triangles. For convenience, on the figure, we have
−−→ −−→
−−→
labeled the vertices OABC and indicated the components of OA, OB and OC.
�
�
��
1
��
5
1 2
�� 1
Area �OAB
=
�
det
=
| − 5| =
.
�
4 3
2
2
2
�
�
�
�
1
�
13
4 3
�� 1
Area �OBC
=
��
det
=
| − 13|
= .
�
3 −1
2
2
2
18
Thus, area of quadrilateral OABC =
= 9.
2
MIT OpenCourseWare
http://ocw.mit.edu
18.02SC Multivariable Calculus
Fall 2010
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