Review on Vectors

Physical Quantities and Units

• Physical Quantities:
  • Made up of numerical values and suitable units.
  • Can be measured with instruments like a meter rule, half-meter rule, measuring tape, vernier calipers, micrometer screw gauge, pendulum clock, watch, stopwatch, and ticker-tape timer.
• Base Quantities:
  • Include length (m), mass (kg), time (s), electric current (A), temperature (K), amount of substance (mol), and luminous intensity (cd).
• Prefixes:
  • micro ($\mu$): $10^{-6}$
  • milli (m): $10^{-3}$
  • centi (c): $10^{-2}$
  • deci (d): $10^{-1}$
  • kilo (k): $10^{3}$
  • mega (M): $10^{6}$
  • May include 'such as' or 'for'.

Scalars

• Scalar Quantities:
  • Described solely by their magnitude.
  • Provide information about how large a measurement is.
  • Examples:
    • Mass: e.g., 14 [kg], 36 [lbs]
    • Time: e.g., 10 seconds, 40 minutes
    • Volume: e.g., 1000 cm3, 4 liters, 12 gallons
    • Temperature: e.g., 14 °F, 25 °C
    • Voltage: e.g., 9 Volts

Operations Involving Scalars

• Scalar quantities (e.g., time, temperature, mass, volume, distance, density, and power) can be added using simple mathematical rules.

Vectors

• Vector Quantities:
  • Described by both magnitude and direction.
  • Provide information about how large a measurement is and the direction of the measurement.
  • Examples:
    • Velocity: e.g., 100 [mi/hr] NORTH
    • Acceleration: e.g., 10 [m/sec$^2$] at 35° with respect to EAST
    • Force: e.g., 980 [Newtons] straight down (270°)
    • Momentum: e.g., 200 [kg m/sec] at 90°

Operations Involving Vectors

• Addition of Vector Quantities
• Multiplication by a Scalar Quantity
• Subtraction of Vector Quantities
• Scalar Product (Dot Product)
• Vector Product (Cross Product)

Graphical Representation of Vectors

• Arrows are used to represent vectors.
• The direction of the arrow relative to some reference point (north or +x axis) gives the direction of the vector.
• The length of a vector arrow is proportional to the magnitude of the vector.
• Example of a displacement vector with a magnitude of 2 km and direction 30° north of east.

Graphical Addition of Vectors

• Put the tail of one vector after the head of the other.

• The resultant vector is an arrow that starts where the first vector starts and ends where the second vector ends.

• If the two vectors are co-linear, especially if they point in the same direction, they add just like scalars.

  • Example: 5 m + 3 m = 8 m

• If the two vectors are not in the same direction, the magnitudes of the vectors can no longer be added like scalars.

  • In this case, if vector A has a magnitude of 27.5 cm due east and vector B has a magnitude of 12.5 cm in a direction 55° north of west, one needs a ruler and a protractor to get a value for the resultant vector.

Adding More Than Two Vectors Graphically

• To add more than two vectors graphically, add them sequentially.
• For example, to add vectors a, b, and c, first add a and b, then add c to the resultant of a and b.
  • a + b + c = (a + b) + c

Addition of Vector by Component Method

• $r<em>x$ and $r</em>y$ are called the components of $\vec{r}$.

• The scalar components $A<em>x$ and $A</em>y$ have the same magnitude as their vector components, but are positive if they point in the positive direction and negative if they point in the negative direction.

• The vector components of $\vec{A}$ are two perpendicular vectors $\vec{A<em>x}$ and $\vec{A</em>y}$ that are parallel to the x and y axes.

• $\vec{A} = \vec{A<em>x} + \vec{A</em>y}$

• The vectors add together using the Pythagorean Theorem so that $A = \sqrt{A<em>x^2 + A</em>y^2}$.

• $A_x = A \cos \theta$

• $A_y = A \sin \theta$

• $\theta = \tan^{-1} (A<em>y / A</em>x)$

Finding the Components of a Vector

• Example: Finding the x- and y-components of an acceleration vector $\vec{a}$ shown in FIGURE 3.17.
• It's important to draw vectors and decompose them into components parallel to the axes.
• Note that the axes are