Alg1131W4T4 - Areas and normals of parallelograms

Friday, 23 March 2018

1:29 PM

Untitled picture.png Machine generated alternative text:
Recall the definition of the cross product of two vectors in R
Find the following cross products Give your answers in Maple vector form < x, y, z >
—4
x
6
Area(OACB)
— a2b1
= lax bl
The area of the parallelogram spanned by a and b is given by the length la x bl . Hence the area of the parallelogram spanned by
4
and
4
sqrt(57)

Untitled picture.png Machine generated alternative text:
3
2
o
3
and 0B
1
E
c
D
x
2
1
The parallelogram OACB spanned by vectors OA
triangles using the diagram above. For example
area(OACB) = IOA x OBI
has an area which can be determined in an elementary way by calculating the areas of rectangles and
2
area(AOAE) = area(ACBH)
and area(AOGB) = area(ACFA)
So since area(ÜODCI)
O we can find that:
area(OACB)
Now reconsider the same parallelogram in three-dimensional space spanned by x, y and z. Then the cross product
OA x 0B
3
1
o
x
1
2
o
which has length 5
O _ So we can see that
Note: make sure to enter your vector in Maple notation, e.g. < 1, 2, 3 >
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Untitled picture.png Machine generated alternative text:
3
2
o
3
and 0B
1
E
c
D
x
2
1
The parallelogram OACB spanned by vectors OA
triangles using the diagram above. For example
area(OACB) = IOA x OBI
has an area which can be determined in an elementary way by calculating the areas of rectangles and
2
area(AOAE) = area(ACBH)
and area(AOGB) = area(ACFA)
So since area(ÜODCI)
O we can find that:
area(OACB)
Now reconsider the same parallelogram in three-dimensional space spanned by x, y and z. Then the cross product
OA x 0B
3
1
o
x
1
2
o
which has length 5
O _ So we can see that
Note: make sure to enter your vector in Maple notation, e.g. < 1, 2, 3 >

Untitled picture.png Machine generated alternative text:
4
—1
i) The area of the parallelogram spanned by the vectors
and
—5
ii) The area of the parallelogram spanned by the general vectors
and
b
Hint: use the absolute value to ensure that the area is a positive number.
iii) The area of the parallelogram spanned by the three-dimensional vectors
Note: the absolute value Izl is input in Maple as abs
in the plane is
20
IS
5
4
-2
abs(ad-öc)
and
3
7
—4
IS
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﷐﷐5﷮4﷮−2﷯﷯×﷐﷐3﷮7﷮−4﷯﷯=﷐﷐−2﷮14﷮23﷯﷯

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Untitled picture.png Machine generated alternative text:
4
—1
i) The area of the parallelogram spanned by the vectors
and
—5
ii) The area of the parallelogram spanned by the general vectors
and
b
Hint: use the absolute value to ensure that the area is a positive number.
iii) The area of the parallelogram spanned by the three-dimensional vectors
Note: the absolute value Izl is input in Maple as abs
in the plane is
20
IS
5
4
-2
abs(ad-öc)
and
3
7
—4
IS

Untitled picture.png Machine generated alternative text:
By providing us with a nifty way of finding a vector perpendicular to two given ones, the cross product is ideally suited to finding normals (i.e. perpendicular vectors) to planes.
i) The plane in three-dimensional space spanned by the vectors
-4
1
2
and
4
6
—2
will have normal vector n
vector n is only determined up to a constant multiple
ii) The parallelogram with adjacent sides AB and AC
where A
= [3, 6,3] , B
—1, —3, 7] and C
—2] has normal vector
O Remember that such a normal
0 and area
equal to sqrt(3946)
Note:
• the length of the vector v
• The Maple notation for the vector
s = which would be input in Maple as sqrt (aA2+bA2+cA2)
1
2
3
is <1,2,3>.

Untitled picture.png Machine generated alternative text:
Consider the unit cube in three-dimensional space, i.e. the cube with vertices
1.5
i) A normal to the plane through the parallelogram OAFG is
ii) The area of the parallelogram OAFG is sqrt(2)
﷐﷐5﷮4﷮−2﷯﷯×﷐﷐3﷮7﷮−4﷯﷯=﷐﷐−2﷮14﷮23﷯﷯
𝐴= ﷐﷮﷐﷐−2﷯﷮2﷯+﷐14﷮2﷯+﷐23﷮2﷯﷯
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Untitled picture.png Machine generated alternative text:
Consider the unit cube in three-dimensional space, i.e. the cube with vertices
1.5
i) A normal to the plane through the parallelogram OAFG is
ii) The area of the parallelogram OAFG is sqrt(2)

Untitled picture.png Machine generated alternative text:
1.5
1.5
iii) A normal to the plane through the triangle AACD is
iv) The area of the triangle AACD is
Note: normals are vectors, so use the Maple notation for vectors

Untitled picture.png Machine generated alternative text:
1.5
1.5
iii) A normal to the plane through the triangle AACD is
iv) The area of the triangle AACD is
Note: normals are vectors, so use the Maple notation for vectors

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