College Physics II – Mechanics, Sound, Oscillations, and Waves
Definition
Vector projection is the process of finding the component of one vector that is parallel to another vector. It represents the length of the projection of one vector onto another vector, and is a fundamental concept in the algebra of vectors.
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The vector projection of a vector $\vec{a}$ onto a vector $\vec{b}$ is given by the formula: $\text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{\|\vec{b}\|^2}\vec{b}$, where $\|\vec{b}\|$ is the magnitude of $\vec{b}$.
Vector projection is a useful concept in physics and engineering, as it allows for the decomposition of vectors into components along specific directions.
The scalar projection of a vector $\vec{a}$ onto a vector $\vec{b}$ is given by the formula: $\text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{\|\vec{b}\|}$.
The vector component of a vector $\vec{a}$ along a vector $\vec{b}$ is given by the formula: $\vec{a}_{\parallel} = \text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{\|\vec{b}\|^2}\vec{b}$.
Vector projection is a linear operation, meaning that $\text{proj}_{\vec{b}}(\vec{a} + \vec{c}) = \text{proj}_{\vec{b}}\vec{a} + \text{proj}_{\vec{b}}\vec{c}$ and $\text{proj}_{\vec{b}}(k\vec{a}) = k\text{proj}_{\vec{b}}\vec{a}$, where $k$ is a scalar.
Review Questions
Explain how the vector projection of a vector $\vec{a}$ onto a vector $\vec{b}$ is calculated, and describe the geometric interpretation of this concept.
The vector projection of a vector $\vec{a}$ onto a vector $\vec{b}$ is calculated using the formula $\text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{\|\vec{b}\|^2}\vec{b}$, where $\|\vec{b}\|$ is the magnitude of $\vec{b}$. Geometrically, this represents the length of the component of $\vec{a}$ that is parallel to $\vec{b}$. The vector projection is the vector that points in the same direction as $\vec{b}$ and has a magnitude equal to the scalar projection of $\vec{a}$ onto $\vec{b}$.
Describe the relationship between the vector projection, scalar projection, and vector component of a vector $\vec{a}$ onto a vector $\vec{b}$, and explain how they are calculated.
The vector projection of a vector $\vec{a}$ onto a vector $\vec{b}$ is given by the formula $\text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{\|\vec{b}\|^2}\vec{b}$. The scalar projection is the magnitude of the vector projection, given by $\text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{\|\vec{b}\|}$. The vector component of $\vec{a}$ along $\vec{b}$ is the vector that represents the projection of $\vec{a}$ onto $\vec{b}$, and is equal to the vector projection: $\vec{a}_{\parallel} = \text{proj}_{\vec{b}}\vec{a}$. These concepts are all related and provide different ways of understanding the relationship between two vectors.
Explain how the properties of linearity apply to vector projection, and discuss the significance of these properties in the context of vector algebra.
Vector projection is a linear operation, meaning that $\text{proj}_{\vec{b}}(\vec{a} + \vec{c}) = \text{proj}_{\vec{b}}\vec{a} + \text{proj}_{\vec{b}}\vec{c}$ and $\text{proj}_{\vec{b}}(k\vec{a}) = k\text{proj}_{\vec{b}}\vec{a}$, where $k$ is a scalar. These properties of linearity are important in vector algebra because they allow for the decomposition of vectors into components along specific directions, and the manipulation of these components using scalar multiplication and vector addition. This facilitates the analysis of vector quantities in physics and engineering, where the ability to break down and recombine vectors is essential for solving problems and understanding the relationships between different vector quantities.
The dot product, or scalar product, of two vectors is a scalar value that is equal to the product of the magnitudes of the two vectors and the cosine of the angle between them.
Scalar Projection: The scalar projection of a vector $\vec{a}$ onto a vector $\vec{b}$ is the scalar value that represents the length of the projection of $\vec{a}$ onto $\vec{b}$.
Vector Component: The vector component of a vector $\vec{a}$ along a vector $\vec{b}$ is the vector that represents the projection of $\vec{a}$ onto $\vec{b}$.