Vectors and Position in Physics with Calculus

Vectors and Coordinate Systems

Representing Position with Vectors

In physics, the position of an object in space can be described using a position vector. This vector indicates both the direction and the distance from the origin to the object's location.

• Position Vector \( \vec{r} \): The position of an object can be written as \( \vec{r} = x \hat{i} + y \hat{j} \), where \( x \) and \( y \) are the coordinates along the x- and y-axes, and \( \hat{i} \), \( \hat{j} \) are unit vectors in those directions.

• Vector Notation: \( \vec{r} = r_x \hat{i} + r_y \hat{j} \)

Vector Addition

Vectors can be added together to find resultant vectors. This is done by adding their corresponding components.

• Component Addition: If \( \vec{A} = A_x \hat{i} + A_y \hat{j} \) and \( \vec{B} = B_x \hat{i} + B_y \hat{j} \), then:

$\vec{A} + \vec{B} = (A_x + B_x) \hat{i} + (A_y + B_y) \hat{j}$

• Graphical Representation: Vectors are often represented as arrows in a coordinate system, and their sum can be visualized using the "tip-to-tail" method.

Scalars and Vectors

In addition to vectors, which have both magnitude and direction, physics also uses scalars. Scalars are quantities that have only magnitude (such as time, mass, and speed).

• Examples of Scalars: Time, mass, speed, temperature.

• Examples of Vectors: Displacement, velocity, force.

Describing Motion in Two Dimensions

When describing the motion of an object in two dimensions, both the x and y components of its position may change with time. The position vector as a function of time is written as:

$\vec{r}(t) = x(t) \hat{i} + y(t) \hat{j}$

• This allows us to track the object's motion in a plane, accounting for changes in both directions.