Medical Physics 2 - Vector Algebra Flashcards

This set of vocabulary flashcards covers fundamental vector algebra concepts, including types of vectors, coordinate systems, products (dot and cross), and different types of fields in physics.

Vector

A physical quantity that possesses both magnitude (size or numerical value) and direction.

Scalar

A physical quantity that has magnitude only and no direction.

Modulus

The magnitude of a vector $\vec{A}$, written as $|\vec{A}|$ or simply $A$.

Equal vectors

Two vectors that have the same magnitude ($|\vec{A}| = |\vec{B}|$) and point in the same direction.

Co-planar vectors

Vectors which are confirmed to the same plane.

Negative vector

A vector with the same magnitude as a given vector but pointing in the opposite direction.

Unit vector

A dimensionless vector that has a magnitude of exactly one and points in a particular direction, defined as $\hat{n} = \frac{\vec{A}}{|\vec{A}|}$.

Position vector

A vector representing the position of a point relative to an arbitrary origin.

Zero vector (Null vector)

Any vector whose magnitude is zero, represented by $\vec{O}$.

Head-to-tail Method

A graphical method to add vectors by placing the tail of the second vector at the head of the first and drawing the resultant from the first tail to the second head.

Commutative Law

A property of vector addition stating that changing the order of addition does not change the resultant: $\vec{a} + \vec{b} = \vec{b} + \vec{a}$.

Associative Law

A property of vector addition stating that grouping does not affect the sum: $\vec{a} + (\vec{b} + \vec{c}) = (\vec{a} + \vec{b}) + \vec{c}$.

Cartesian Coordinate System

A coordinate system consisting of three mutually perpendicular axes: $x$-axis, $y$-axis, and $z$-axis.

Three-dimensional Magnitude Formula

The magnitude of a vector calculated using the expression $a = \sqrt{a_x^2 + a_y^2 + a_z^2}$ based on Pythagoras' theorem.

Standard Unit Vectors

Three mutually perpendicular unit vectors in the Cartesian system: $\hat{i}$ (positive $x$-axis), $\hat{j}$ (positive $y$-axis), and $\hat{k}$ (positive $z$-axis).

Polar Coordinates

A two-dimensional system describing a point using radial distance ($r$) from the origin and angular position ($\theta$) from the positive $x$-axis.

Position-dependent unit vectors

Unit vectors such as $\hat{r}$ and $\hat{\theta}$ in polar coordinates that change direction from one point to another.

Scalar (Dot) Product

An operation on two vectors that produces a scalar quantity, defined as $\vec{A} \cdot \vec{B} = AB\cos(\theta)$, where $\theta$ is the angle between them.

Dot Product Cartesian Form

The algebraic calculation of the scalar product using components: $\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z$.

Vector (Cross) Product

An operation combining two vectors to produce another vector perpendicular to both, defined as $\vec{A} \times \vec{B} = AB\sin(\theta)\hat{n}$.

Right-Hand Rule

A rule used to determine the direction of a cross product: point fingers along $\vec{A}$, rotate them toward $\vec{B}$, and the thumb points in the direction of $\vec{A} \times \vec{B}$.

Anticommutative Property

A property of the cross product where reversing the order of the operation changes the sign: $\vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})$.

Vector Field

A mathematical function $\vec{F}(x, y, z)$ that associates one vector with every point in space, which can vary from point to point.

Scalar Field

A mathematical function $U(x, y, z)$ that assigns a single numerical value (magnitude only) to every point in space, such as temperature or pressure.