Glossary

Vector products are crucial in physics, allowing us to describe complex interactions between forces and objects. They come in two flavors: scalar products (dot products) and vector products (cross products), each with unique properties and applications.

These mathematical tools help us calculate work, , and . Understanding vector products is key to grasping mechanics, as they provide a powerful way to analyze forces and motion in three-dimensional space.

Vector Products

Scalar vs vector products

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  • () yields a scalar quantity denoted by A⃗⋅B⃗\vec{A} \cdot \vec{B} and calculated as ∣A⃗∣∣B⃗∣cos⁡θ|\vec{A}||\vec{B}|\cos\theta where θ\theta is the angle between vectors
    • Commutative property: A⃗⋅B⃗=B⃗⋅A⃗\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}
    • Determines work done by a force W=F⃗⋅d⃗W = \vec{F} \cdot \vec{d} and of one vector onto another
  • () yields a vector quantity denoted by A⃗×B⃗\vec{A} \times \vec{B} with ∣A⃗∣∣B⃗∣sin⁡θ|\vec{A}||\vec{B}|\sin\theta
    • perpendicular to the plane containing A⃗\vec{A} and B⃗\vec{B} determined by the
    • Anti-commutative property: A⃗×B⃗=−(B⃗×A⃗)\vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})
    • Calculates torque τ⃗=r⃗×F⃗\vec{\tau} = \vec{r} \times \vec{F}, L⃗=r⃗×p⃗\vec{L} = \vec{r} \times \vec{p}, and magnetic force on a moving charge F⃗=qv⃗×B⃗\vec{F} = q\vec{v} \times \vec{B}

Vector algebra

  • of a vector represents its length or size
  • Direction of a vector indicates the orientation in space
  • is used for vector addition
  • applies: A⃗⋅(B⃗+C⃗)=A⃗⋅B⃗+A⃗⋅C⃗\vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}

Calculation of scalar products

  • Calculate using vector components: A⃗⋅B⃗=AxBx+AyBy+AzBz\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z
  • Calculate using magnitudes and angle between vectors: A⃗⋅B⃗=∣A⃗∣∣B⃗∣cos⁡θ\vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}|\cos\theta
  • Positive indicates force has a component in the same direction as displacement (work done)
  • Negative scalar product indicates force has a component opposite to displacement direction (work against)
  • Zero scalar product means force is perpendicular to displacement resulting in no work done
  • Determines the component of A⃗\vec{A} along the direction of B⃗\vec{B} through projection ∣A⃗∣cos⁡θ|\vec{A}|\cos\theta

Computation of vector products

  • Calculate using vector components: A⃗×B⃗=(AyBz−AzBy)i^−(AxBz−AzBx)j^+(AxBy−AyBx)k^\vec{A} \times \vec{B} = (A_yB_z - A_zB_y)\hat{i} - (A_xB_z - A_zB_x)\hat{j} + (A_xB_y - A_yB_x)\hat{k}
  • Calculate using the determinant of a 3x3 matrix: A⃗×B⃗=∣i^j^k^AxAyAzBxByBz∣\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}
  • Magnitude ∣A⃗×B⃗∣=∣A⃗∣∣B⃗∣sin⁡θ|\vec{A} \times \vec{B}| = |\vec{A}||\vec{B}|\sin\theta represents the area of the parallelogram formed by A⃗\vec{A} and B⃗\vec{B}
  • Direction perpendicular to the plane containing A⃗\vec{A} and B⃗\vec{B} determined by the right-hand rule (point fingers along A⃗\vec{A}, curl towards B⃗\vec{B}, thumb points in direction of A⃗×B⃗\vec{A} \times \vec{B})

Vector products in mechanics

  • Torque calculated as τ⃗=r⃗×F⃗\vec{\tau} = \vec{r} \times \vec{F} where r⃗\vec{r} is position vector from axis of rotation to force application point and F⃗\vec{F} is the applied force
    • Magnitude ∣τ⃗∣=∣r⃗∣∣F⃗∣sin⁡θ|\vec{\tau}| = |\vec{r}||\vec{F}|\sin\theta where θ\theta is angle between r⃗\vec{r} and F⃗\vec{F}
    • Direction perpendicular to plane containing r⃗\vec{r} and F⃗\vec{F} determined by right-hand rule
  • Angular momentum calculated as L⃗=r⃗×p⃗\vec{L} = \vec{r} \times \vec{p} where r⃗\vec{r} is position vector from origin to particle and p⃗\vec{p} is linear momentum of particle
    • Magnitude ∣L⃗∣=∣r⃗∣∣p⃗∣sin⁡θ|\vec{L}| = |\vec{r}||\vec{p}|\sin\theta where θ\theta is angle between r⃗\vec{r} and p⃗\vec{p}
    • Direction perpendicular to plane containing r⃗\vec{r} and p⃗\vec{p} determined by right-hand rule
  • Conservation of angular momentum: total angular momentum of a system remains constant in the absence of external torques

Key Terms to Review (31)

×: The multiplication symbol, also known as the cross symbol, is a mathematical operation that represents the product of two or more numbers or quantities. It is a fundamental operation in vector mathematics, particularly in the context of vector products.
Angle between two vectors: The angle between two vectors is the measure of the smallest rotation required to align one vector with the other. It can be calculated using the dot product and magnitudes of the vectors.
Angular momentum: Angular momentum is a measure of the quantity of rotation of an object and is a vector quantity. It is given by the product of the moment of inertia and angular velocity.
Angular Momentum: Angular momentum is a fundamental concept in physics that describes the rotational motion of an object. It is the measure of an object's rotational inertia and its tendency to continue rotating around a specific axis. Angular momentum is a vector quantity, meaning it has both magnitude and direction, and it plays a crucial role in understanding the behavior of rotating systems across various topics in physics.
Anticommutative: Anticommutativity is a property of certain mathematical operations, such as the cross product of vectors, where the order of the operands affects the result. In other words, the operation is not commutative, meaning that $a \times b \neq b \times a$.
Anticommutative property: The anticommutative property states that the order of the operands affects the sign of the result. Specifically, for vectors, $\mathbf{a} \times \mathbf{b} = - (\mathbf{b} \times \mathbf{a})$.
Corkscrew right-hand rule: The corkscrew right-hand rule is a mnemonic used to determine the direction of the cross product vector in three-dimensional space. Point your right-hand thumb in the direction of the first vector and curl your fingers towards the second vector; your thumb points in the direction of the resulting vector.
Cross product: The cross product is a binary operation on two vectors in three-dimensional space, resulting in another vector that is perpendicular to the plane containing the original vectors. It is denoted by $\mathbf{A} \times \mathbf{B}$ and has both magnitude and direction.
Cross Product: The cross product, also known as the vector product, is a binary operation on two vectors that results in a third vector that is perpendicular to both of the original vectors. It is a fundamental concept in vector algebra and has important applications in various fields of physics, such as electromagnetism and mechanics.
Direction: Direction refers to the orientation of a vector in space, indicating where it is pointing relative to a reference point or coordinate system. It is crucial in understanding how vectors represent physical quantities like displacement, velocity, and acceleration, as each of these requires both magnitude and direction for a complete description.
Distributive Property: The distributive property is a fundamental algebraic rule that states the product of a number and a sum is equal to the sum of the individual products. It allows for the simplification of mathematical expressions by distributing one factor to the individual terms within a sum or difference.
Dot product: The dot product is a scalar quantity obtained by multiplying the magnitudes of two vectors and the cosine of the angle between them. It provides a measure of how much one vector extends in the direction of another.
Dot Product: The dot product, also known as the scalar product, is a mathematical operation that combines two vectors to produce a scalar quantity. It is a fundamental concept in vector algebra that has important applications in various areas of physics, including the study of scalars, vectors, and their interactions.
Magnitude: Magnitude is the size or length of a vector, representing its absolute value. It is always a non-negative scalar quantity.
Magnitude: Magnitude is a quantitative measure that describes the size, scale, or extent of a physical quantity. It is a fundamental concept in physics that is essential for understanding and analyzing various physical phenomena.
Newton-Meters: Newton-meters (N⋅m) are the units used to measure torque, which is the rotational force that causes an object to rotate around a fixed point or axis. Torque is a vector quantity that describes the tendency of a force to cause rotational motion.
Orthogonality: Orthogonality is a fundamental concept in linear algebra and vector analysis, describing the property of two or more vectors being perpendicular or at right angles to each other. This term is particularly relevant in the context of vector products, as it helps define the relationships between different vector operations.
Parallelogram rule: The parallelogram rule is a geometric method used to determine the resultant of two vectors by forming a parallelogram with the two vectors as adjacent sides. This rule highlights how vectors can be combined to find a single vector that represents their combined effect, making it crucial for understanding vector addition and the properties of vector spaces.
Products of Vectors: The product of two vectors is a new vector that is derived from the original vectors. This operation allows for the calculation of quantities such as work, torque, and magnetic force, which are essential in understanding the behavior of physical systems.
Projection: Projection is the process of representing a three-dimensional object or space onto a two-dimensional surface, such as a screen or a piece of paper. It involves mapping the spatial coordinates of the object or space onto a flat plane while preserving certain geometric properties.
Right-Hand Rule: The right-hand rule is a mnemonic device used to determine the direction of various vector quantities, such as the cross product of two vectors, the direction of torque, angular momentum, and the precession of a gyroscope. It provides a simple and intuitive way to visualize and remember the orientation of these physical quantities.
Scalar product: The scalar product, also known as the dot product, is a mathematical operation that takes two vectors and returns a single scalar. It is calculated as the product of the magnitudes of the two vectors and the cosine of the angle between them.
Scalar Product: The scalar product, also known as the dot product, is a fundamental operation in vector mathematics that combines two vectors to produce a scalar quantity. It is a way of multiplying vectors to obtain a single number, which represents the projection of one vector onto the other.
Torque: Torque is a measure of the rotational force applied to an object, which causes it to rotate about an axis. It is influenced by the magnitude of the force applied, the distance from the axis of rotation, and the angle at which the force is applied, making it crucial for understanding rotational motion and equilibrium.
Unit Vector: A unit vector is a dimensionless vector with a magnitude of 1 that points in a specific direction. It is used to represent the direction of a vector without regard to its magnitude.
Unit vectors of the axes: Unit vectors of the axes are vectors that have a magnitude of 1 and point in the direction of the coordinate axes. They are typically denoted as $\hat{i}$, $\hat{j}$, and $\hat{k}$ in three-dimensional space for the x, y, and z-axes respectively.
Vector Algebra: Vector algebra is a branch of mathematics that deals with the operations and properties of vectors, which are mathematical entities that have both magnitude and direction. It provides a systematic way to perform various operations on vectors, such as addition, subtraction, scalar multiplication, and vector multiplication, to analyze and solve problems in physics, engineering, and other scientific fields.
Vector product: The vector product, also known as the cross product, is a binary operation on two vectors in three-dimensional space. It results in a third vector that is perpendicular to the plane of the original vectors and has a magnitude equal to the area of the parallelogram they span.
Vector Product: The vector product, also known as the cross product, is a binary operation on two vectors that results in a third vector that is perpendicular to both of the original vectors. The vector product is an important concept in physics and mathematics, particularly in the context of studying the relationships between different physical quantities represented by vectors.
Work-energy theorem: The work-energy theorem states that the net work done on an object is equal to its change in kinetic energy. Mathematically, it is expressed as $W_{net} = \Delta KE$.
Work-Energy Theorem: The work-energy theorem is a fundamental principle in physics that states the change in the kinetic energy of an object is equal to the net work done on that object. It establishes a direct relationship between the work performed on an object and the resulting change in its kinetic energy, providing a powerful tool for analyzing and solving problems involving energy transformations.
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