11.3
The dot product combines two vectors and produces a scalar value.
Algebraically, it is calculated by multiplying the corresponding components of the vectors and adding the products. Geometrically, it equals the product of the magnitudes of both vectors and the cosine of the angle between them.
A practical application is optimizing the tilt of solar panels for maximum energy absorption. Sunlight is modeled as a three-dimensional direction vector.
Another vector represents the panel’s normal direction, perpendicular to its surface. Since sunlight travels toward the panel, its vector points inward. To align with the outward normal, the sunlight vector is reversed, and the dot product is calculated between the panel normal and the reversed vector.
This means multiplying the corresponding components and adding the results. Dividing this product by the vector magnitudes gives the cosine term that controls energy absorption by measuring how directly sunlight hits the panel.
When the angle is small, the cosine is large, meaning greater energy absorption.
At 90 degrees, the dot product becomes zero, and the panel receives no direct solar energy.
The dot product, or scalar product, is a fundamental operation in vector algebra that combines two vectors to yield a scalar quantity. It is particularly valuable in physical applications, such as calculating work, and in mathematical contexts, such as determining vector projections and direction cosines.
For vectors a = ⟨a₁, a₂, a₃⟩ and b = ⟨b₁, b₂, b₃⟩, the dot product is defined as:
\begin{equation*}\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\end{equation*}
This operation is commutative and distributive over vector addition and is compatible with scalar multiplication. If vectors a and b are orthogonal, then their dot product equals zero. When vectors have equal direction, the dot product is maximized for their magnitudes.
The dot product also has a geometric definition involving the angle θ between two vectors:
\begin{equation*}\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta\end{equation*}
This formulation determines the angle between vectors or checks for orthogonality (θ = 90° implies a·b = 0).
The angles between a vector and the positive coordinate axes are known as direction angles. If α, β, and γ are the angles between vector a and the x-, y-, and z-axes, respectively, the direction cosines are:
\begin{equation*}\cos \alpha = \frac{a_1}{|\mathbf{a}|}, \quad \cos \beta = \frac{a_2}{|\mathbf{a}|}, \quad \cos \gamma = \frac{a_3}{|\mathbf{a}|}\end{equation*}
These satisfy the identity:
\begin{equation*}\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\end{equation*}
The scalar projection of b onto a (compₐ b) and the vector projection (projₐ b) extend compₐ b by scaling the unit vector in the direction of a:
\begin{align*}\text{comp}_{\mathbf{a}} \mathbf{b} &= \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}|}\\\text{proj}_{\mathbf{a}} \mathbf{b}& = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}|^2} \mathbf{a}\end{align*}
These projections describe how much of b aligns with a, with physical significance in mechanics and electromagnetism.
The dot product combines two vectors and produces a scalar value.
Algebraically, it is calculated by multiplying the corresponding components of the vectors and adding the products. Geometrically, it equals the product of the magnitudes of both vectors and the cosine of the angle between them.
A practical application is optimizing the tilt of solar panels for maximum energy absorption. Sunlight is modeled as a three-dimensional direction vector.
Another vector represents the panel’s normal direction, perpendicular to its surface. Since sunlight travels toward the panel, its vector points inward. To align with the outward normal, the sunlight vector is reversed, and the dot product is calculated between the panel normal and the reversed vector.
This means multiplying the corresponding components and adding the results. Dividing this product by the vector magnitudes gives the cosine term that controls energy absorption by measuring how directly sunlight hits the panel.
When the angle is small, the cosine is large, meaning greater energy absorption.
At 90 degrees, the dot product becomes zero, and the panel receives no direct solar energy.
From Chapter 11:
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