Vectors and scalars are fundamental concepts in physics. Vectors have both magnitude and direction, like velocity, while scalars only have magnitude, like speed. Understanding these helps us describe motion and forces accurately.

Coordinate systems provide a framework for measuring position and motion. One-dimensional systems use a single axis, with positive and negative directions. This allows us to track an object's position, displacement, and direction of movement.

Vectors and Scalars

Scalar vs vector quantities

• Scalar quantities characterized by magnitude only no direction (mass, temperature, time)
• Vector quantities have both magnitude and direction (displacement, velocity, acceleration)
• Scalar examples include speed which is the magnitude of velocity, energy a measure of an object's capacity to do work
• Vector examples include force which has both magnitude and direction, momentum the product of mass and velocity

Coordinate systems for motion

• One-dimensional motion occurs along a straight line described using a single coordinate axis
  • Positive direction typically chosen as right or upward, objects moving this way have positive displacement, velocity, acceleration
  • Negative direction is opposite, objects moving this way have negative displacement, velocity, acceleration
• Origin is the reference point where position is zero
• Position denoted by variable $x$ is location of object relative to origin
• Displacement $\Delta x$ is change in position, positive when moving in positive direction, negative in negative direction

Vector notation in physics

• Vectors represented with arrow above symbol ($\vec{v}$ for velocity, $\vec{a}$ for acceleration, $\vec{F}$ for force)
• Magnitude is absolute value of vector ($|\vec{v}|$ for speed, $|\vec{a}|$ for acceleration magnitude, $|\vec{F}|$ for force magnitude)
• Direction specified by angle or relative to coordinate system ($\vec{v} = 5 \text{ m/s, east}$, $\vec{F} = 10 \text{ N, 30° above horizontal}$)
• Vectors can be decomposed into components along coordinate axes ($\vec{F} = F_x \hat{i} + F_y \hat{j}$)
  • $\hat{i}$ and $\hat{j}$ are unit vectors in $x$ and $y$ directions
• Vector addition involves combining multiple vectors to determine a resultant vector

Vector components and coordinate systems

• Cartesian coordinates use perpendicular axes (x, y, z) to describe vector components
• Polar coordinates use magnitude and angle to describe vectors, useful for circular motion
• Vector components are projections of a vector onto coordinate axes
• Magnitude of a vector can be calculated from its components using the Pythagorean theorem

Coordinate Systems

Coordinate systems for motion

• One-dimensional motion occurs along straight line, described with single coordinate axis
  • Positive direction usually right or upward
  • Negative direction is opposite
• Origin is reference point where position is zero
• Position $x$ is object's location relative to origin
• Displacement $\Delta x$ is change in position
  • Positive when moving in positive direction
  • Negative when moving in negative direction