The Mystery of the Minus Sign: Finding the Cross Product of a x b

In summary, the cross product of two vectors is a vector that is perpendicular to both vectors and has a magnitude equal to the product of their magnitudes multiplied by the sine of the angle between them. It can be calculated using a specific formula and a negative value signifies opposite directions. The magnitude of the cross product can also be used to find the area of a parallelogram. In real life, the cross product is used in various fields such as physics, engineering, and computer graphics to calculate torque, magnetic fields, determine object orientation, and create 3D effects and animations.
  • #1
tony873004
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find the cross product a x b of the vectors a = 5.0m,i and b=4.3m,k
[tex]5.0m\ast 4.3m\ast \sin (90)=22m^2,\hat {j}[/tex]

The back of the book gives
[tex]-(22m^2)\hat {j}[/tex]

Where does the minus sign come from?
 
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  • #2
Naturally

[tex] \vec{i}\times\vec{k}=-\vec{j} [/tex]

,because

[tex]\vec{k}\times\vec{i}=\vec{j} [/tex]

Daniel.
 
  • #3
thanks, dextercioby
 
  • #4
Or: "right hand rule". Curl the fingers of your right hand from the positive x-axis (i) to the positive z-axis "j". You thumb will be pointing down the negative y-axis.
 

Related to The Mystery of the Minus Sign: Finding the Cross Product of a x b

1. What is the cross product of two vectors?

The cross product of two vectors, a and b, is a vector that is perpendicular to both a and b and has a magnitude equal to the product of the magnitudes of a and b multiplied by the sine of the angle between them.

2. How do you calculate the cross product of two vectors?

The cross product of two vectors, a and b, can be calculated using the following formula:
a x b = (aybz - azby)i + (azbx - axbz)j + (axby - aybx)k

3. What does a negative cross product value signify?

A negative cross product value signifies that the resulting vector is pointing in the opposite direction of the positive value. This can also be interpreted as the two vectors being in the opposite direction of each other.

4. Can you use the cross product to find the area of a parallelogram?

Yes, the magnitude of the cross product of two vectors is equal to the area of the parallelogram formed by those two vectors.

5. How is the cross product used in real life applications?

The cross product has many real life applications, including in physics, engineering, and computer graphics. It is used to calculate torque, magnetic fields, and determine the orientation of objects in 3D space. It is also used in computer graphics to create 3D effects and animations.

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