Dot Product and Its Properties

Dot Product

• The dot product is an operation that multiplies corresponding components of two vectors and adds the products together. This operation results in a scalar, not a vector.
• To find the dot product of two vectors A and B, you would add the products of their corresponding components: A1 * B1 + A2 * B2 + A3 * B3
• The dot product is commutative, meaning the order in which you multiply the vectors doesn't affect the result.
• The dot product is distributive over vector addition.
• The dot product is associative with scalar multiplication.
• Dotting a vector with the zero vector will result in zero.
• Dotting a vector with itself (V * V) produces the square of the magnitude of the vector (||V||^2).

Dot Product Properties & Applications

• Alternate form: The dot product of two vectors can be expressed as the product of their magnitudes and the cosine of the angle between them: V • W = ||V|| * ||W|| * cos(θ).
• Finding the angle between two vectors: You can solve for the angle (θ) between two vectors using the dot product: θ = arccos((V • W) / (||V|| * ||W||)).
• Orthogonality: If the dot product of two vectors is zero, then the vectors are perpendicular (also known as orthogonal or normal).
• Mutually Orthogonal Vectors: The standard basis vectors, i, j, and k, are mutually orthogonal. This means that the dot product of any two of them is always zero.

Key Concepts & Examples

• The dot product can be used to determine if two vectors are parallel or perpendicular.
• If the angle between two vectors is 0, they are parallel.
• If the angle between two vectors is π, they are parallel and in opposite directions.
• If the angle between two vectors is π/2, they are perpendicular (orthogonal).
• The law of cosines can be used to derive the alternate form of the dot product.
• The dot product is useful for finding the angle between vectors, which has applications in geometry, physics, and other areas where analyzing vectors is important.

Dot Product of Unit Vectors

• The dot product of a unit vector with itself is always 1.
• This is because the angle between a unit vector and itself is 0 degrees.
• The cosine of 0 degrees is 1.

Finding the Angle Between Two Vectors

• The angle between two vectors can be found using the dot product.
• Formula: cos(θ) = (v • w) / (||v|| ||w||)
  • v and w are the two vectors.
  • v • w is the dot product of the vectors.
  • ||v|| and ||w|| are the magnitudes of the vectors.

Parallel Vectors

• Two vectors are parallel if they are scalar multiples of each other.
• To determine if two vectors are parallel, try to factor out a scalar from one vector to obtain the other vector.

Orthogonal Vectors

• Two vectors are orthogonal (perpendicular) if their dot product is zero.
• To determine if two vectors are orthogonal, calculate their dot product. If the result is zero, they are orthogonal.

Parallel and Orthogonal Vectors

• To determine if two vectors are parallel, check if they are scalar multiples of each other.
• To determine if two vectors are orthogonal (perpendicular), calculate their dot product. If the dot product equals zero, the vectors are orthogonal.

Vector Projection

• Vector projection is the idea of finding the component of one vector in the direction of another.
• The projection of vector V onto vector W is denoted as proj_WV.
• The component projection of vector V onto vector W is the magnitude of the vector projection.

Formulas for Component and Vector Projection

• Component Projection:
  • If the angle between V and W is known: comp<sub>W</sub>V = ||V|| cos(θ), where θ is the angle between the vectors.
  • If the angle is unknown: comp<sub>W</sub>V = (V • W) / ||W||
• Vector Projection:
  • proj<sub>W</sub>V = (V • W) / ||W||<sup>2</sup> * W

Key Points About Vector Projection

• The vector being projected onto (W in the formulas) determines the direction of the projection, represented by its unit vector.
• The vector being projected (V in the formulas) determines the magnitude of the projection, represented by its scalar projection (component projection).
• When the angle between two vectors is greater than π/2 (90°), the component projection will be negative, indicating that the projection is in the opposite direction of the vector being projected onto.

Application of Vector Projection

• Vector projection can be used to calculate the amount of force being applied in a specific direction, such as determining how much of a force pulling a dog uphill is actually contributing to the dog's upward movement.
• The formula can be manipulated to determine the angle between two vectors using only the dot product and magnitudes of the vectors.

Vector Projection

• Vector projection is a process of finding the component of a vector that lies along the direction of another vector.

• Formula for vector projection of b onto a:

  • proj_ab = [(a•b)/(||a||²)]a

• The formula uses dot product of two vectors and the magnitude squared of the vector onto which you are projecting.

• The result of the vector projection is a vector that represents the component of b in the direction of a.

• The formula produces a scalar multiplied by a vector, resulting in a vector that has the same direction as a but a magnitude determined by the scalar value.

Understanding Vector Projection

• The vector projection tells us how much of vector b is aligned with vector a.
• This can be interpreted as the part of vector b that is "going in the a direction."
• To find the length of this component, calculate the magnitude of the projected vector.

Work in Physics

• Work is the component of a force vector along a path directed in a certain direction.

• This is essentially a form of vector projection.

• Work is defined as the product of the magnitude of the force vector and the magnitude of the path vector, multiplied by the cosine of the angle between them.

• This aligns with the concept of vector projection because it involves calculating the component of the force vector that is acting in the direction of the path.

• Work, unlike force, is a scalar quantity.

• The work formula: Work = ||F|| ||D||cos(θ)

Scalar and Vector Properties in Work

• Work is scalar, meaning it has magnitude but not direction.

• The dot product involved in the work formula (||F|| ||D||cos(θ)) highlights the commutative property of scalar quantities, meaning that work can be calculated regardless of the order of the force and displacement vectors.

• This commutative property is useful for understanding that the work done is the same regardless of the order of the force and displacement vectors.

• The work formula emphasizes the relationship between vector projection and work as the component of a force vector contributing to displacement.