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Right Handed System of Vectors

When two vectors

u

and

v

are kept tail-to-tail,then placing the right hand along the direction of

u

, and curling the fingers in the direction of the angle

v

makes with

u

, the thumb points in the direction of

u \times v

.
A three-dimensional coordinate system in which the axes satisfy the right-hand rule is called a right-handed coordinate system, while one that does not is called a left-handed coordinate system.

Definition

Cross Product of Unit Vectors

\vec{a} \times \vec{b} = \hat{n}

(For perpendicular vectors)
(where

\hat{n}

is a unit vector perpendicular to both the vectors)

Example

Angle Between Vectors using cross-product

Angle between vectors can be determined using cross-product by:

s i n (θ) = \frac{| \vec{a} \times \vec{b} |}{| a | | b |}

Formula

Area of triangle and parallelogram using cross product

The cross product of

\vec{P}

and

\vec{Q}

can found out as follows:

\vec{P} \times \vec{Q}

\hat{i} (P_{y} Q_{z} - P_{z} Q_{y}) + \hat{j} (P_{z} Q_{x} - P_{x} Q_{z}) + \hat{k} (P_{x} Q_{y} - P_{y} Q_{x})

The magnitude of cross product of two vectors is numerically equal to the area of parallelogram whose adjacent sides represent the two vectors.
In the fig.