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Cross product and perpendicular component

Settings

1. a vector and a plane perpendicular to 2. a vector 3. divide into two components : the component vector of parallel to (i.e. ) : the component vector of perpendicular to (i.e. ) Observe that .

Try with your right hand

4. Align your right-hand in the direction of . 5. Curl towards . 6. Observe the direction of your right thumb. That's the direction of the cross product . We denote the cross product (or vector product) of vectors and as .

Explore 3D

  1. Click ToolsOthersRotate 3D Graphics/Move Graphics to change viewing angle.
  2. Scale the image with the middle scroll button of your mouse or two fingers on your touchpad.
  3. If necessary, use full screen by clicking the button on the bottom right corner.
  4. Click Show calculations and observe the bolded text. What do you see?

3D animation

Key Observations

1. The magnitude of is . 2. The direction of : is perpendicular to the plane containing both and . Rotate your right little finger from to anticlockwise. + If the angle is between 0° and 180°, then you should follow the direction of your right thumb. + Otherwise, you should take the inverse direction of your right thumb. 3. , where is the component of perpendicular to . (i.e. )