04. Iloczyn wektorowy - ang. (Vector product)

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Vector product


In the space additionally to the scalar product of two vectors, which is a scalar size, there is another kind of vector multiplication. This new useful definition of the two vector products defines a new vector.

Vector product of vectors \vec{a} and \vec{b} is defined as

  \vec{a}\times \vec{b}=\vec{n}|a||b|\sin \alpha


where
         
 |a|, |b| are the lengths of vectors \vec{a}, \vec{b};
•           \alpha is the angle (smaller from  180 ^ \circ ) between  \vec {a} and attached at one point;
         vector  \vec {n} is a unit vector perpendicular to the plane defined by  \vec {a} and  \vec {b} (as shown close to the hand).

 
 


Figure 1: Illustration of the vector product of vectors  \vec {a} and  \vec {b} by the right hand rule.

 
The right hand rule:
In the vector product vectors have the following orientation:

•  vector \vec {a} as the index finger

•  vector  \vec {b} as the middle finger

•  vector \vec {c} and the vector  \vec {n} as  the thumb

It can also be interpreted as:

• The direction  \vec {n} we determine by the motion of screw in the clockwise direction , if we turn it from the vector  \vec {a} to the vector \vec {b} along the smaller angle between them.

• winding a small hand on the big one on the face of the clock the vector product will have a perpendicular direction to the face of a clock facing us from the wall.

• winding (unlike before) a big hand on  the small one  on the face of the clock then the vector product will have a perpendicular direction to the face of a clock facing us to the wall.

Vector product can be illustrated as follows: [http://demonstrations.wolfram.com/CrossProductOfVectors/ ­­­-
simulation in the format of cdf
]

Features:

• Since  \sin 0 = 0 where  \alpha = 0 , we have that:

                                 \Vec {a} \times \vec {a} = 0

We can even say more:

 Vector product of parallel vectors is zero.

• When we change the order of winding, the thumb must be pointed down, or

 \vec {b} \times \vec {a} = - \vec {a} \times \vec {b}.

• Vector multiplication of axis vector is convenient  \vec {i} ,  \vec {j} ,  \vec{k} , which can be stored in the table
 

  

 Read the table from ‘the left to the right", for example:

 \vec {i} \times \vec {j} = \vec {k} or  \vec {i} \times\vec{k} =-\vec {j}.

• Vector product has the property of separation relatively to  addition:

 \vec {a} \times \left (\vec {b} +\vec {c} \right) = \vec {a} \times \vec {b}+\vec {a} \times \vec {c}


Example

\vec{i}\times\left(\vec{j}+\vec{k}\right)=\vec{i}\times\vec{j}+\vec{i}\times\vec{k}=\vec{k}-\vec{j}=[0,-1,1],

Note: Try to draw it.

• Formula for calculating the coordinates of the vector product is known, when we have the

coordinates of the vectors \vec{a}=[a_x,a_y,a_z] ,\vec{b}= [b_x,b_y,b_z].Then:

\vec{a}\times\vec{b}=\vec{i}\left|\begin{array}{cc} a_y & a_z \\ b_y & b_z \end{array}\right| -\vec{j}\left|\begin{array}{cc} a_x & a_z \\ b_x & b_z\end{array}\right| +\vec{k}\left|\begin{array}{cc} a_x & a_y \\ b_x & b_y\end{array}\right|,

where the determinants of a second degree we count as follows:

          \[\left|\begin{array}{cc}a & b\\c & d\end{array}\right |=ad-bc

 

Example
We count  \vec {u} \times \vec {v} for  \vec {u} = [3, -1,0] ,  \vec {v} = [0, 2, 3].

Solution:
With this formula we get:

 \vec {u} \times \vec {v} = \vec {i} \left | \begin {array} {rr} -1 & 0 \\2 & 3 \end {array} \right | - \vec {j} \left | \begin {array} {rr} 3 & 0 \\0 & 3 \end {array} \right | + \vec {k} \left | \begin {array} {rr} 3 & -1 \\0 & 2 \end {array} \right | = -3 \vec {i} -9 \vec {j}+ 6 \vec {k} = [-3, -9.6].


Applications
Vector product has a lot of useful interpretation both physical and geometric. We are mentioning some of them below:

 

• the area of the parallelogram in the picture of the hand is equal to the length of the vector product   P = | \vec {a} \times \vec {b} |. Also on the area of ​​a triangle built on vectors  \vec {a} ,  \vec {b} can be calculated as:  P _ {\Delta} = \frac {1} {2} | \vec {a} \times \vec {b} |.

• Using the concept of vector product we can define various physical quantities, such as angular momentum, torque, and also recorded a number of rights of mechanics and electrodynamics.

  

Example
Calculate the area of ​​a parallelogram, whose three vertices are the points of  O = (0,0,0) ,  A = (1,1,1) ,  B = (2,3,5) .
Solution:
Parallelogram is ‘unfasten’ by the vectors:  \vec{OA} = [1,1,1] ,  \vec {OB} = [2,3,5] . Then

\vec{OA}\times \vec{OB}=\vec{i}\left|\begin{array}{rr} 1 & 1 \\ 3 & 5 \end{array}\right| -\vec{j}\left|\begin{array}{rr} 1 & 1 \\ 2 & 5 \end{array}\right|+\vec{k}\left|\begin{array}{rr} 1 & 1 \\ 2 & 3 \end{array}\right|=2\vec{i}-3\vec{j}+\vec{k}=[2,-3,1]

 

P=\left| \vec{OA}\times \vec{OB}\right|=\sqrt{2^2+(-3)^2+1^2}=\sqrt{14}.

 

 

Figure 2: Illustration for the field parallelogram, whose three vertices are the points

 O = (0, 0, 0), ,  A = (1, 1, 1) ,  B = (2, 3, 5)


Interesting information and notes

Another interpretation: If the points of attachment of three vectors cover each other, then the observer located in the plane spanned by the vectors  \vec {a} and  \vec {b} looking in the direction of the vector  \vec { c} , can pass along the shortest path from the direction of the vector  \vec {a} to the direction \vec {b} by doing the turn opposite to clockwise direction.

Note: Please note that the rule applies to right-handed orientation. It can be accepted left handed rule, then we go clockwise and use the left hand rule.

• Right hand or left hand rule is sometimes called Fleming in honor of the British

physicist John Ambrose Fleming, who applied the rule in the study of electromagnetism

in the late nineteenth century.

• The vector product has many technological uses. It is used for example in determining the

Lorentz force, this is the force which acts on the electric charge in the electromagnetic field,

which is a system of two fields: the electric and magnetic fields. Since the force acts always transverse on the moving fraction relative to the velocity vector and vector induction, so

 overall induction vector can be defined by the vector product. Thus, vector product can

be found in the equations defining the transformation of electric and magnetic fields in the

transition to a moving system.

• To calculate at some point a magnetic induction produced by any distribution of currents we divide each of the current into infinitesimal elements and calculate the contribution from each component and then we add up them and get the vector of magnetic induction as the resultant vector. The contribution of each current element is given by the Biot-Savart law, which is used in electromagnetism and fluid dynamics. In the formula known today as the Biot –Savart - Laplace law vector product appears.

• Articles may be used during mathematics and physics lessons in high schools

when discussing vector calculus and its applications.

Reminder

The vector unit ( versor)  \vec {e} is called a vector whose length is 1. Each

vector  \vec {a} can be represented as the product of the unit vector and its length.

• When we divide (non-zero) vector by the length, we get a unit vector:

 \vec {e} = \frac {\vec {a}} {| a |}}.

• To describe the coordinate axes in three-dimensional space is often used versor axes: {tex}

\vec {i} = [1,0,0] {/tex},  \vec {j} = [0,1,0] ,  \vec {k} = [0,0,1] .

Each vector $ \vec {a} $ can be distributed into a sum of three versors multiplied by the

the vector coordinates  \vec {a} = [a_x, a_y, a_z] :

 \vec {a} = a_x \vec {i} + a_y \vec {j} + a_z \vec {k}}.

For example,  [1,2,3] = \vec {i}+2 \vec {j}+3 \vec {k} .

• The length of the vector  \vec {a} = [x, y, z] is calculated using the formula:

 | \vec {a} | = \sqrt {x ^ 2 + y ^ 2 + z ^ 2} .
 

CHECK WHAT YOU KNOW
Underline the correct answer.
1.Vector  product is:

the number

vector
unit vector.

2. Vector product in the right-hand coordinate system we set according to the rule of:

right hand
left hand
the right foot.

3.Vector product  \left (\vec {k} +2 \vec {i} \right) \times \vec {i} is     

 - \vec {i}

 0

 \vec {j} .
4.Vector product  \vec {i} \times \left (\vec {k} + \vec {j} \right) is

 - \vec {i}

 \vec {k} - \vec {j}

 \vec {j} - \vec {k}
 

 

Note to the student:

1.  If you do not remember the previous topic read the reminder of the needed information.

2.  Look carefully at theFigure 1, analyze the vectors position, practice on your own hand.

3.  Then turn applets with attached links:

        http://www.phy.syr.edu/courses/java-suite/crosspro.html

        http://demonstrations.wolfram.com/CrossProductOfVectors

4.  Practice calculating the vector product of the formula before proceeding.
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Teacher's notes:

1.  Tell students to read the required reminder messages.

2.  Figure 1 should be discussed with the students, to analyze the position vectors, work out

     on your own hand.

3. Then turn the applets with accompanying links: http://www.phy.syr.edu/courses/java

    suite/crosspro.html http://demonstrations.wolfram.com/CrossProductOfVectors/

4.  Practice with the students to calculate the  vector product of the formula before        proceeding.

5.  Practice with the students how to calculate the volume of solids entering the coordinate system.