Section 4.1

Areas, Volumes, and Cross Products

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The Area of a Parallelogram

A parallelogram determined by two non-zero and non-parallel vectors a =[ , ] and b = [ , ] in .

> plotparellelogram();

>

What is the area of this parallelogram?

Area of the parallelogram is given as,

(1) Area = (base)(altitude) = ||a|| h = ||a|| ||b|| sin( ) = ||a|| ||b|| .

Note: (page 24) (a b) = ||a|| ||b|| .

Square both sides of the equation (1) to get,

= ( ) = - = - .

Notice:

=

=

=

Thus, we have

= + + =

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Area of the parallelogram formed by two non-zero and non-parallel vectors

a =[ , ] and b = [ , ] in is given by Area =| |. The

number inside the absolute value signs is known as the determinant of the matrix,

det(A) = .

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Example 1: Find the area of the parallelogram given, where

>

> a := matrix(2,1,[3,1.5]); b := matrix(2,1,[1,2]);

> plotparellelogram2();

> A := augment(a,b);

> `Area` = det(A);

>

Cross Products

Given two independent vectors v = [ ] and u= [ ] in . We want to find a vector

p = [ ] in that is perpendicular to both vectors v and u .

> transpose(matrix(3,1,[p[1],p[2],p[3]])) * matrix(3,1,[u[1],u[2],u[3]]) = 0; transpose(matrix(3,1,[p[1],p[2],p[3]])) * matrix(3,1,[v[1],v[2],v[3]]) = 0;

>

> evalm(transpose(matrix(3,1,[p[1],p[2],p[3]])) &* matrix(3,1,[u[1],u[2],u[3]])) = 0; evalm(transpose(matrix(3,1,[p[1],p[2],p[3]])) &* matrix(3,1,[v[1],v[2],v[3]])) = 0;

>

> matrix(2,3,[v[1],v[2],v[3],u[1],u[2],u[3]]) * matrix(3,1,[p[1],p[2],p[3]])= 0 ;

>

Find the nullspace.

> matrix(2,4,[v[1],v[2],v[3],0,u[1],u[2],u[3],0]);

> AUG := rref(matrix(2,4,[v[1],v[2],v[3],0,u[1],u[2],u[3],0]));

> p[1] = (v[3]*u[2] - v[2]*u[3])/(u[1]*v[2]-v[1]*u[2]) * p[3]; p[2] = (u[1]*v[3] - u[3]*v[1])/(u[1]*v[2]-v[1]*u[2]); p[3] = p[3];

>

Let, = . Then we have,

> p[1] = v[3]*u[2] - v[2]*u[3]; p[2] = u[1]*v[3] - u[3]*v[1]; p[3] =v[2]*u[1]-v[1]*u[2];

> p = det(matrix(2,2,[u[2],u[3],v[2],v[3]]))*i - det(matrix(2,2,[u[1],u[3],v[1],v[3]]))*j + det(matrix(2,2,[u[1],u[2],v[1],v[2]]))*k;

Where i=[1,0,0], j=[0,1,0] and k=[0,0,1].

> p = Det(matrix(2,2,[u[2],u[3],v[2],v[3]]))*i - Det(matrix(2,2,[u[1],u[3],v[1],v[3]]))*j + Det(matrix(2,2,[u[1],u[2],v[1],v[2]]))*k;

>

The vector p is known as the cross product of the two vectors u and v , i.e. p = u x v.

Use the determinat of the symbolic matrix ,

> p = Det(matrix(3,3,[i,j,k,u[1],u[2],u[3],v[1],v[2],v[3]]));

>

Example 2: Find a vector perpendicular to,

> v := matrix(3,1,[-1, 3, -2]); u :=matrix(3,1,[3, -2, 1]);

> v1 := [-1, 3, -2]: u1 := [3, -2, 1]:

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> A := transpose(augment(matrix(3,1,[i,j,k]),u,v));

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> `u x v` = det(A);

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> A := transpose(augment(matrix(3,1,[i,j,k]),v,u));

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> `v x u` = det(A);

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Right hand Rule

> plotcrossproduct(v1,u1);

>

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Let v = [ ] and u= [ ] in . Then the area of the parallelogram in

formed by u and v is given as || u x v || or || v x u ||.

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Example 3: Find the area of the parallelogram formed by the vectors,

> v := matrix(3,1,[1,4,6]); u :=matrix(3,1,[6,3,1]);

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> A := transpose(augment(matrix(3,1,[i,j,k]),u,v));

>

> `u x v` = det(A); p := det(A):

>

> `||u x v||` = sqrt(coeff(p,i)^2 + coeff(p,j)^2 + coeff(p,k)^2);

>

Example 1 (revisted): Find the area of the parallelogram given, where

>

> a := matrix(3,1,[3,1.5,0]); b := matrix(3,1,[1,2,0]);

>

> A := transpose(augment(matrix(3,1,[i,j,k]),a,b));

>

> `a x b` = det(A); p := det(A):

>

> `||a x b||` = sqrt(coeff(p,i)^2 + coeff(p,j)^2 + coeff(p,k)^2);

>

Properties of the cross product

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Let a and b be vectors in .

1.) b x c = -(c x b)

2.) a x (b x c) is generally different from (a x b) x c

3.) a x (b+c) = a x b + a x c

(a+b) x c = (a x c) + (b x c)

4.) b (b x c) = (b x c) c = 0 Since (b x c) is perpendicular to b and c.

5.) || b x c || = Area of the parallelogram determined by b and c.

6.) a ( b x c) = (a x b) c = ( + or -) Volume of the box determined by a, b, and c.

7.) a x (b x c) = (a c) b - (a b) c

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Exercise

1, 3, 5-7, 10-15, 19-21, 25, 26, 30-34.