Vectors and Coordinate Systems Lecture Notes

Vocabulary and key formulas covering coordinate systems, vector properties, and vector addition from the Chapter 3 lecture transcript.

Cartesian Coordinate System

A system where x- and y- axes intersect at the origin and points are labeled $(x,y)$.

Polar Coordinate System

A coordinate system where a point is distance $r$ from the origin in the direction of angle $\theta$ measured counterclockwise from a reference line, labeled $(r, \theta)$.

Distance

A scalar quantity representing the total path length traveled by a particle.

Displacement

A vector quantity representing the solid line from point A to B; it is independent of the path taken between these points.

Equality of Two Vectors

Two vectors are equal if they possess the same magnitude and point in the same direction along parallel lines.

Resultant

The vector drawn from the origin of the first vector to the end of the last vector in a tip-to-tail graphical addition.

Commutative Law of Addition

A rule stating that when two vectors are added, the sum is independent of the order of addition: $\mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A}$.

Associative Property of Addition

A rule stating that when adding three or more vectors, the sum is independent of the grouping: $(\mathbf{A} + \mathbf{B}) + \mathbf{C} = \mathbf{A} + (\mathbf{B} + \mathbf{C})$.

Negative of a Vector

A vector that has the same magnitude as the original but points in the opposite direction; adding it to the original vector results in zero.

Component

A projection of a vector along an axis, such as the x- or y-axis, used to completely describe the vector.

Unit Vector

A dimensionless vector with a magnitude of exactly $1$ used to specify a direction with no other physical significance.

Position Vector

A vector used to specify a point in the xy plane, typically written in unit-vector notation as $\mathbf{r} = xî + yĵ$.

$$r = \sqrt{x^2 + y^2}$$

The formula used to convert Cartesian coordinates to the magnitude component of polar coordinates.

$$\theta = \tan^{-1}(\frac{y}{x})$$

The formula used to find the angle component of polar coordinates from Cartesian coordinates.

$A_x = A \cos(\theta)$ and $A_y = A \sin(\theta)$

The formulas for finding the rectangular components of a vector $\mathbf{A}$.