Physics - Vectors and Scalars

Physicist's Goal

  • Precisely define fundamental measurable quantities.

  • Find relationships between these quantities.

  • Need vectors and scalars to achieve these goals.

Scalars

  • Quantities with magnitude only.

  • Examples:

    • Mass

    • Distance

    • Speed

    • Volume

    • Temperature

    • Energy

Vectors

  • Quantities with both magnitude and direction.

  • More complex to work with compared to scalars.

Vector Notation

  • Arrow over the symbol indicates a vector.

  • Example: v⃗\vec{v} for velocity.

  • Typed vector quantities are sometimes written in bold.

    • v (velocity, vector)

    • v (speed, scalar).

  • In one-dimensional scenarios, directions are indicated by positive or negative signs.

  • Coordinate system:

    • Right: positive direction

    • Left: negative direction.

Drawing Vectors

  • Arrow represents a vector visually.

  • Length: magnitude.

  • Direction: direction of the quantity.

  • Example:

    • Velocity of 5 m/s to the right.

    • Velocity of 10 m/s to the left.

  • Vectors should be drawn to scale relative to each other.

  • Tip/Head: pointed end of the arrow.

  • Tail: the other end of the arrow.

  • A vector can be moved in space as long as its magnitude and direction remain constant.

One-Dimensional Coordinate System

  • Horizontal dimension:

    • Right: positive.

    • Left: negative.

  • A vector can be expressed in different ways:

    • v⃗=10ms left\vec{v} = 10 \frac{m}{s} \text{ left}

    • v⃗=−10ms\vec{v} = -10 \frac{m}{s}

Adding Vectors Visually

  • Tip-to-tail method:

    • Place the tail of the second vector at the tip of the first vector.

    • Draw a new vector from the tail of the first vector to the tip of the last vector.

    • This new vector is the resultant (sum) of the vectors.

  • Order of addition does not matter.

  • Works for any number of vectors in one, two, or three dimensions.

Subtracting Vectors Visually

  • Subtracting a vector is equivalent to adding its negative.

  • The negative of a vector has the same magnitude but points in the opposite direction.

  • Example: Subtracting x<em>2⃗\vec{x<em>2} from x</em>1⃗\vec{x</em>1}

    • Add −x<em>2⃗\vec{-x<em>2} to x</em>1⃗\vec{x</em>1}

Adding/Subtracting Vectors Numerically

  • Find the magnitude of the resultant vector without drawings.

  • Example:

    • x1⃗=5 m to the right\vec{x_1} = 5 \text{ m to the right}

    • x2⃗=8 m to the left\vec{x_2} = 8 \text{ m to the left}

  • Coordinate system:

    • Left: negative direction

    • Right: positive direction

  • Assign signs:

    • x1⃗=5 m\vec{x_1} = 5 \text{ m}

    • x2⃗=−8 m\vec{x_2} = -8 \text{ m}

  • R=x<em>1+x</em>2=5 m+(−8 m)=−3 mR = x<em>1 + x</em>2 = 5 \text{ m} + (-8 \text{ m}) = -3 \text{ m}

Problem Solving Tips

  • Do not add vectors that represent different types of quantities.

    • Example: Do not add temperature to volume.

    • Example: Do not add displacement to velocity.

  • Pay attention to units.

  • Clearly define your coordinate system.

  • Remember you are working with vectors, even if direction (sign) is implied.