Dot & Cross Product Operations

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Introduction

The dot product and cross product are two fundamental operations in vector algebra. Although they both involve multiplying vectors, they produce very different results.
The dot product gives a scalar that measures how aligned two vectors are, while the cross product gives a vector perpendicular to both original vectors and measures the area they span.

Dot Product (Scalar Product)

Geometric Definition

For two vectors \vec{a} and \vec{b}, the dot product is defined as:

\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}| \cos\theta

where \theta is the angle between the vectors.

Component Form

If \vec{a} = (a_x, a_y, a_z) and \vec{b} = (b_x, b_y, b_z), then:

\vec{a}\cdot\vec{b} = a_x b_x + a_y b_y + a_z b_z

Angle Between Two Vectors

\cos\theta = \frac{\vec{a}\cdot\vec{b}}{|\vec{a}|,|\vec{b}|}

If
ā€¢ \vec{a}\cdot\vec{b} > 0 ā†’ angle is acute
ā€¢ \vec{a}\cdot\vec{b} < 0 ā†’ angle is obtuse
ā€¢ \vec{a}\cdot\vec{b} = 0 ā†’ vectors are orthogonal

Projection of One Vector on Another

Scalar projection:

\text{proj}_{\vec{b}}(\vec{a}) = \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}

Vector projection:

\text{Proj}_{\vec{b}}(\vec{a}) = \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|^2}\vec{b}

Physical Interpretation

ā€¢ Work done: W = \vec{F}\cdot\vec{d}
ā€¢ Power: P = \vec{F}\cdot\vec{v}
ā€¢ Detecting perpendicular directions in mechanics and navigation

Cross Product (Vector Product)

Geometric Definition

|\vec{a} \times \vec{b}| = |\vec{a}|,|\vec{b}| \sin\theta

The direction is perpendicular to the plane containing both vectors (right-hand rule).

Determinant (Standard Representation)

\vec{a}\times\vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}

Expanded Component Form

\vec{a}\times\vec{b} = \begin{aligned} &(a_y b_z - a_z b_y),\hat{i}\ &-(a_x b_z - a_z b_x),\hat{j}\ &+(a_x b_y - a_y b_x),\hat{k} \end{aligned}

Geometric & Physical Meaning

Area of the parallelogram formed by the two vectors:

A = |\vec{a} \times \vec{b}|

Area of triangle:

A = \frac{1}{2}|\vec{a}\times\vec{b}|

Cross product gives directions for several physical quantities (e.g., torque, angular momentum).

Properties

\vec{a}\times\vec{b} = -(\vec{b}\times\vec{a})
\vec{a}\times(\vec{b}+\vec{c}) = \vec{a}\times\vec{b} + \vec{a}\times\vec{c}
Zero if vectors are parallel.

Scalar Triple Product

\vec{a}\cdot(\vec{b}\times\vec{c}) = \begin{vmatrix} a_x & a_y & a_z \\ b_x & b_y & b_z \\ c_x & c_y & c_z \end{vmatrix}

Gives the volume of the parallelepiped formed by the three vectors.
If the value is zero, the vectors are coplanar.

Vector Triple Product

\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a}\cdot\vec{c})\vec{b} + (\vec{a}\cdot\vec{b})\vec{c}

Useful in simplifying expressions in mechanics, electromagnetics, and fluid dynamics.

Applications in Engineering

Torque:

\vec{\tau} = \vec{r} \times \vec{F}

Angular momentum:

\vec{L} = \vec{r} \times \vec{p}

Magnetic force:

\vec{F} = q(\vec{v}\times\vec{B})

These operations are extensively used in aerospace dynamics, 3D rotations, structural analysis, and electromagnetics.

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