In this page, the vector cross product is explained in first principles.

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The product where vector components in perpendicular interact is defined. Given vectors `vec p` and `vec q` as shown in figure. What is the component of `vec q` perpendicular to the `vec p`?

- `vec a`
- `vec b`
- `vec b`
- `ai+bj`
- `bj+ai`

The answer is '`vec b`'

Given `vec q` with component `vec b` in perpendicular to `vec p` as shown in figure. The angle between the vectors is `theta`. What is `|vec b|`?

- `|q|sin theta`
- `|q|sin theta`
- `|q|cos theta`
- `|p|sin theta`
- `|p|cos theta`

The answer is '`|q|sin theta`'

Given vectors `vec p` and `vec q` and the product between them as shown in the figure. What will be a good choice of direction of the product between components in perpendicular?

Two intersecting lines, which are not parallel, define a plane. The normal on the face of the plane describes the plane and so, the normal is taken as the direction of the product.

A plane can be described by

• normal which is one side of the plane, or

• the negative of the normal, which is the other side of the plane. Which one is chosen for the vector cross product `vec p xx vec q`?

The direction of the cross product `vec p xx vec q`. In a standard right-handed rectangular coordinate system,

The direction in which a standard screw advances when it turns from `vec p` to `vec q`, defines the normal.

The direction of the cross product `vec p xx vec q`. In a standard right-handed rectangular coordinate system,

The direction pointed by the right thumb when fingers are curled to point from `vec p` to `vec q`, defines the normal.

Formal definition of **vector cross product**.

`vec p xx vec q = |p||q|sin theta hat n` where `hat n` is the normal denoting the direction of right handed rotation from `vec p` to `vec q`.

**Vector Cross Product ** is defined as the product of components in perpendicular.

**Vector Cross Product: ** for vectors `vec p, vec q in bbb(R)^3`

`vec p xx vec q = (|p||q|sin theta ) hat n`

where `hat n` is the unit vector of right-handed rotation from `vec p` to `vec q`.

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