Dot product or scalar product of two vectors

Twelve Standard >> Dot product or scalar product of two vectors

Dot Product or Scalar Product of Two Vectors

Vectors are essential tools in mathematics and physics, representing quantities that possess both magnitude and direction. Among the different operations that can be performed on vectors, the dot product (also called the scalar product) is fundamental for understanding projections, angles, and work done by a force.

What is Scalar Product?

The scalar product of two vectors is an algebraic operation that takes two vectors and returns a single number (a scalar). Unlike vector multiplication that results in another vector, the scalar product results in a real number.

Mathematically, the dot product of two vectors A and B is defined as:

\( \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos \theta \)

Where:

\(|\mathbf{A}|\) and \(|\mathbf{B}|\) denote the lengths or magnitudes of the vectors \(\mathbf{A}\) and \(\mathbf{B}\), respectively.
\(\theta\) is the angle between the two vectors

Algebraically, if \(\mathbf{A} = A_1\hat{i} + A_2\hat{j} + A_3\hat{k}\) and \(\mathbf{B} = B_1\hat{i} + B_2\hat{j} + B_3\hat{k}\), then:

\( \mathbf{A} \cdot \mathbf{B} = A_1B_1 + A_2B_2 + A_3B_3 \)

What is Vector Product?

The vector product or cross product of two vectors results in a vector that is perpendicular to both of the original vectors. It is denoted by \(\mathbf{A} \times \mathbf{B}\).

Mathematically:

\( \mathbf{A} \times \mathbf{B} = |\mathbf{A}||\mathbf{B}|\sin\theta\ \hat{n} \)

Where:

\(\hat{n}\) represents a unit vector that is orthogonal to both \(\mathbf{A}\) and \(\mathbf{B}\), with its direction identified using the right-hand rule.

Important Properties and Results on Dot Product

Commutative Property: \( \mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A} \)
Distributive over addition: \( \mathbf{A} \cdot (\mathbf{B} + \mathbf{C}) = \mathbf{A} \cdot \mathbf{B} + \mathbf{A} \cdot \mathbf{C} \)
Orthogonality: If \( \mathbf{A} \cdot \mathbf{B} = 0 \), then vectors \( \mathbf{A} \) and \( \mathbf{B} \) are perpendicular.
Self Dot Product: \( \mathbf{A} \cdot \mathbf{A} = |\mathbf{A}|^2 \)

Geometrical Interpretation

The dot product helps determine the extent to which one vector aligns with the direction of another. It is particularly useful for finding the angle between two vectors using the formula:

\( \cos \theta = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}||\mathbf{B}|} \)

Cross Product of Unit Vectors

Cross products of unit vectors are particularly useful in 3D space calculations and are defined as follows:

Cross Product	Result
\( \hat{i} \times \hat{j} \)	\( \hat{k} \)
\( \hat{j} \times \hat{k} \)	\( \hat{i} \)
\( \hat{k} \times \hat{i} \)	\( \hat{j} \)
\( \hat{j} \times \hat{i} \)	\( -\hat{k} \)
\( \hat{k} \times \hat{j} \)	\( -\hat{i} \)
\( \hat{i} \times \hat{k} \)	\( -\hat{j} \)
\( \hat{i} \times \hat{i},\ \hat{j} \times \hat{j},\ \hat{k} \times \hat{k} \)	\( 0 \)

Applications of Dot Product

Work done: \( W = \mathbf{F} \cdot \mathbf{d} = |\mathbf{F}||\mathbf{d}|\cos\theta \)
Projection of a vector: The projection of \( \mathbf{A} \) on \( \mathbf{B} \) is \( \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{B}|} \)
Finding angle between vectors

Conclusion

The dot product (or scalar product) is a crucial operation in vector algebra, especially when analyzing direction, angle, and projection. Understanding both dot and cross products enhances one’s ability to work with physical quantities like force, displacement, and motion in three-dimensional space.