Cross product and perpendicular component
- Author:
- Vincent Tam
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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
- Click Tools → Others → Rotate 3D Graphics/Move Graphics to change viewing angle.
- Scale the image with the middle scroll button of your mouse or two fingers on your touchpad.
- If necessary, use full screen by clicking the button on the bottom right corner.
- 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. )