14.2 Dot Product and Projections

The dot product is a powerful tool in vector math, letting us multiply vectors to get a single number. It's super useful for finding angles between vectors and projecting one vector onto another. These operations come in handy all the time in physics and engineering.

Vector decomposition using the dot product helps us break down forces and motions into their parts. This is key in many real-world problems, from analyzing forces on structures to understanding how objects move in 3D space.

Vector Operations

Dot Product and Scalar Product

Top images from around the web for Dot Product and Scalar Product

• 

  Vectors and the Geometry of Space | Boundless Calculus View original

  Is this image relevant?

• 

  Dot product - Wikipedia View original

  Is this image relevant?

• 

  Dot product - Wikipedia View original

  Is this image relevant?

• 

  Vectors and the Geometry of Space | Boundless Calculus View original

  Is this image relevant?

• 

  Dot product - Wikipedia View original

  Is this image relevant?

1 of 3

Top images from around the web for Dot Product and Scalar Product

• 

  Vectors and the Geometry of Space | Boundless Calculus View original

  Is this image relevant?

• 

  Dot product - Wikipedia View original

  Is this image relevant?

• 

  Dot product - Wikipedia View original

  Is this image relevant?

• 

  Vectors and the Geometry of Space | Boundless Calculus View original

  Is this image relevant?

• 

  Dot product - Wikipedia View original

  Is this image relevant?

1 of 3

• The dot product, also known as the scalar product, is a binary operation that takes two vectors and returns a scalar
• Denoted by $\vec{a} \cdot \vec{b}$ or $\langle \vec{a}, \vec{b} \rangle$, where $\vec{a}$ and $\vec{b}$ are vectors
• Algebraically, the dot product is calculated by multiplying the corresponding components of the vectors and then summing the products: $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + \ldots + a_nb_n$
• The dot product is commutative: $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$
• The dot product is distributive over vector addition: $\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$

Vector Decomposition using Dot Product

• The dot product can be used to decompose a vector into components parallel and perpendicular to another vector
• Given vectors $\vec{a}$ and $\vec{b}$, the projection of $\vec{a}$ onto $\vec{b}$ is given by: $\text{proj}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{\|\vec{b}\|^2} \vec{b}$
  • This formula gives the component of $\vec{a}$ parallel to $\vec{b}$
• The component of $\vec{a}$ perpendicular to $\vec{b}$ can be found by subtracting the projection from the original vector: $\vec{a}_{\perp} = \vec{a} - \text{proj}_{\vec{b}} \vec{a}$
• Vector decomposition is useful in physics and engineering applications where forces or velocities need to be resolved into components

Geometric Interpretations

Projection and Angle between Vectors

• Geometrically, the dot product represents the projection of one vector onto another
• The projection of $\vec{a}$ onto $\vec{b}$ is the length of the component of $\vec{a}$ parallel to $\vec{b}$, with a sign depending on the angle between the vectors
• The dot product is related to the angle $\theta$ between two vectors by the formula: $\vec{a} \cdot \vec{b} = \|\vec{a}\| \|\vec{b}\| \cos(\theta)$
  • This formula is known as the geometric definition of the dot product
• The angle between two vectors can be calculated using the inverse cosine: $\theta = \arccos\left(\frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\| \|\vec{b}\|}\right)$

Orthogonal Vectors and Zero Dot Product

• Two vectors are orthogonal (perpendicular) if their dot product is zero: $\vec{a} \cdot \vec{b} = 0$
• Orthogonal vectors have a 90° angle between them, as $\cos(90°) = 0$
• The zero vector is orthogonal to all vectors, as the dot product of the zero vector with any vector is always zero
• Orthogonal vectors are important in many applications, such as defining coordinate systems and constructing orthonormal bases

Applications

Work and Dot Product

• In physics, work is defined as the product of force and displacement in the direction of the force
• The dot product is used to calculate work when the force and displacement vectors are not parallel
• Given a force vector $\vec{F}$ and a displacement vector $\vec{d}$, the work done by the force is: $W = \vec{F} \cdot \vec{d} = \|\vec{F}\| \|\vec{d}\| \cos(\theta)$
  • $\theta$ is the angle between the force and displacement vectors
• If the force is perpendicular to the displacement ($\theta = 90°$), no work is done, as $\cos(90°) = 0$
• The dot product allows for the calculation of work in more general situations where the force and displacement vectors are not necessarily parallel or perpendicular

Other Applications of Dot Product

• The dot product is used in computer graphics to calculate lighting and shading effects by determining the angle between surface normals and light sources
• In machine learning, the dot product is used to measure the similarity between feature vectors, which is essential for algorithms like k-nearest neighbors and support vector machines
• The dot product is also used in signal processing to calculate the correlation between two signals, which helps in pattern recognition and signal filtering applications