332 Scalar Projection

A scalar measuring how much of one vector lies along another that is used to extract the parallel part of a vector.

definition (Scalar Projection) The signed magnitude of the vector projection of \(\mathbf{b}\) onto \(\mathbf{a}\), also referred to as the component of \(\mathbf{b}\) along \(\mathbf{a}\), denoted by \(\text{comp}_{\mathbf{a}} \mathbf{b}\). It represents the length of the “shadow” and is positive if the angle between the vectors is acute and negative if it is obtuse. It is defined by the following formulas:

• \(\text{comp}_{\mathbf{a}} \mathbf{b} = \dfrac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}|}\)
• \(\text{comp}_{\mathbf{a}} \mathbf{b} = |\mathbf{b}| \cos \theta\)

where

• \(\mathbf{a}, \mathbf{b}\) are vectors in a coordinate system.
• \(\theta\) is the angle between vectors \(\mathbf{a}\) and \(\mathbf{b}\).
• \(|\mathbf{b}|\) is the magnitude of vector \(\mathbf{b}\).

• Basic Derivation of Jacobian
• Chain Rule
• Cross Product
• Curl
• Curve
• Definite Integral
• Derivation of Curl
• Derivation of Divergence
• Derivation of Grad
• Derivation of Jacobian
• Derivation of Taylor Expansion
• Divergence
• Dot Product
• Envelope of Tangent Lines
• General Functions
• Gradient
• Improper Integral
• Indefinite Integral
• Jacobian
• Legendre Transform
• Legendre Transform Derivation
• Limit
• Line Integral
• Notes
• Parameterization
• Partial Derivative
• Real Parameter
• Scalar Projection
• Smooth Curve
• Vector Calculus
• Vector Differential Operator
• Vector Field
• Vector Projection