Products of Vectors
Introduction
Vector products allow us to combine vectors in different ways, depending on whether we need a scalar result (dot product) or a vector result (cross product). These operations are essential in physics and engineering for understanding quantities like work, torque, and angular momentum.
Dot Product (Scalar Product)
The dot product of two vectors and
gives a scalar value. It is defined as:
Here:
: Magnitude of
.
: Magnitude of
.
: Angle between
and
.
If and
are expressed in components, the dot product is:
Example: If and
:
The dot product is .
Cross Product (Vector Product)
The cross product of two vectors and
gives a vector that is perpendicular to both
and
. It is defined as:
Here:
and
: Magnitudes of
and
.
: Angle between
and
.
: Unit vector perpendicular to both
and
, determined by the right-hand rule.
If and
are expressed in components, the cross product is calculated using the determinant:
Example: If and
:
Expanding the determinant:
The cross product is .
Applications of Vector Products
- Dot Product: Used to calculate work done (
) and projection of vectors.
- Cross Product: Used to calculate torque (
) and angular momentum.
Key Takeaways
Dot products result in scalars, while cross products result in vectors. Both operations are crucial for analyzing physical phenomena, such as forces, motion, and energy.
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