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|2.5||Vector Product of Two Vectors|
In these revision notes for Vector Product of Two Vectors, we cover the following key points:
Symbolically, the cross product of two vectors a⃗ and b⃗ is denoted through the symbol (×).
Geometrically, it represents a new vector, which is perpendicular to the plane on which the two vectors lie.
Mathematically, the cross product of two vectors represents the magnitude of the surface area enclosed by the two vectors a⃗ and b⃗ and their parallel extensions (the area of the parallelogram formed by the two vectors a⃗ and b⃗).
The cross product of two vectors a⃗ and b⃗ is
and its magnitude is
where θ is the angle between the vectors a⃗ and b⃗.
For the cross product of two vectors, the following rule is true
If the coordinates of the vectors a⃗ and b⃗ (namely xa, ya, za, xb, yb and zb) are given, we can find the coordinates of the vector c⃗ = a⃗ × b⃗ (i.e. xc, yc and zc) using the following formulae:
In Physics, there are many applications of vectors cross product. Some of them include:
If we multiply the cross product of two vectors with a scalar, the result is still a vector as the cross product gives a vector and the product of a vector by a scalar gives a vector as well.
In the cross product
we can apply the "right hand rule" to find the direction of the vector product c⃗ when the directions of a⃗ and b⃗ are known. Thus, the index and middle fingers represent the vectors a⃗ and b⃗ respectively, while the thumb shows the direction of vector product
based on the drill (screwdriver) rule.
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