Lec 03: Vectors | 8.01 Classical Mechanics, Fall 1999 (Walter Lewin)

Main ideas & lessons (with structure)

1) Scalars vs. vectors in physics

• Scalars are quantities determined uniquely by one number (examples):
  • Mass
  • Temperature
  • Speed
• Vectors require both magnitude and direction:
  • Velocity is a vector (speed + direction)
  • Acceleration is a vector
• Vectors are represented by arrows:
  • Arrow length = magnitude
  • Arrow direction = direction in space
• Visual note about representation:
  • “Looking head-on” vs. “from behind” shows dot vs. cross symbols (used later for vector notation conventions).

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2) How to add vectors (the “secret behind adding vectors”)

• Key physical/mathematical idea:
  • If two motions happen simultaneously, the observed displacement equals the vector sum.

• Relationship used to define addition: [ \vec{OS} = \vec{OP} + \vec{PS} ]

• Two primary vector addition methods:

Head-tail technique

• Place the tail of vector B at the head of vector A.
• The resultant is (\vec{C} = \vec{A} + \vec{B}).
• Order doesn’t matter: [ \vec{A}+\vec{B} = \vec{B}+\vec{A} ]

Parallelogram method

• Bring the tails of (\vec{A}) and (\vec{B}) together.
• Complete the parallelogram.
• The diagonal gives (\vec{C} = \vec{A} + \vec{B}).
• Also shows commutativity: [ \vec{A}+\vec{B} = \vec{B}+\vec{A} ]

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3) Negative vectors and vector subtraction

• Negative vector definition: [ \vec{A} - \vec{A} = \vec{0} ] To add to (\vec{A}) and get zero, you must add (-\vec{A}).

• Geometric meaning:

  • (-\vec{A}) is the same vector flipped by 180°.

• Vector subtraction rewritten as addition: [ \vec{A} - \vec{B} = \vec{A} + (-\vec{B}) ]

• Equivalent viewpoint:

  • If (\vec{A} - \vec{B} = \vec{C}), then (\vec{C} + \vec{B} = \vec{A}).
• Practical advice:
  • The head-tail method is described as the easiest and safest for subtraction too.

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4) Vector decomposition (breaking a vector into components)

• Since a vector can be represented as a single resultant from multiple others, you can reverse the idea:
  • Decomposition = replacing one vector by a sum of simpler directional pieces.
• In 3D, decompose (\vec{A}) along the x-, y-, and z-axes using unit vectors:
  • Unit vectors point along positive axes and have magnitude 1:
    • (\hat{x}), (\hat{y}), (\hat{z})

• Component form: [ \vec{A} = A_x \hat{x} + A_y \hat{y} + A_z \hat{z} ]

• Magnitude formula: [ |\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2} ]

• Example used:

  • (\vec{A} = 3\hat{x} - 5\hat{y} + 6\hat{z})
  • Magnitude: [ |\vec{A}| = \sqrt{3^2 + (-5)^2 + 6^2} = \sqrt{70} ]

• Angle relationships (via a cosine relation): [ \cos\theta = \frac{A_z}{|\vec{A}|} ] Example: [ \cos\theta = \frac{6}{\sqrt{70}} ]

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5) Vector multiplication: dot product and cross product

5.1 Dot product (scalar product)

• Definition: result is a scalar (no direction).

• Component definition: [ \vec{A}\cdot\vec{B} = A_xB_x + A_yB_y + A_zB_z ]

• Geometric definition (when the angle (\theta) between vectors is known): [ \vec{A}\cdot\vec{B} = |\vec{A}|\,|\vec{B}|\,\cos\theta ]

• Sign behavior based on (\theta):

  • If (\theta = 90^\circ) (perpendicular): (\cos\theta=0) ⇒ dot product = 0
  • If (0^\circ<\theta<90^\circ): (\cos\theta>0) ⇒ dot product positive
  • If (90^\circ<\theta<180^\circ): (\cos\theta<0) ⇒ dot product negative
• Physics connection preview:
  • Work and energy involve dot products, so sign matters (positive vs. negative work).
• Simple example calculations were mentioned (e.g., cases leading to a negative result; (\hat{y}\cdot\hat{z}=0)).

5.2 Cross product (vector product)

• Definition: [ \vec{A}\times\vec{B} = \vec{C} ]

  • Result is a vector perpendicular to both (\vec{A}) and (\vec{B}).
  • Two taught methods:

Method 1: determinant/matrix “recipe” (always works)

• Set up:
  • Row 1: (\hat{x}, \hat{y}, \hat{z})
  • Row 2: (A_x, A_y, A_z)
  • Row 3: (B_x, B_y, B_z)

• Component-by-component: [ C_x = A_yB_z - A_zB_y ] [ C_y = A_zB_x - A_xB_z ] [ C_z = A_xB_y - A_yB_x ]

• Assemble: [ \vec{C} = C_x\hat{x}+C_y\hat{y}+C_z\hat{z} ]

Method 2: geometric (magnitude via sine + direction via corkscrew rule)

• Magnitude: [ |\vec{A}\times\vec{B}| = |\vec{A}|\,|\vec{B}|\,\sin\theta ]

• Zero conditions:

  • (\theta=0^\circ) or (180^\circ) ⇒ parallel/anti-parallel ⇒ magnitude 0
• Direction rule:
  • Rotate from (\vec{A}) to (\vec{B}) through the shortest angle
  • Use the right-hand corkscrew rule:
    • Into the page vs. out of the page depends on the order of vectors.

• Antisymmetry: [ \vec{A}\times\vec{B} = -(\vec{B}\times\vec{A}) ]

• Contrast with dot product symmetry: [ \vec{A}\cdot\vec{B} = \vec{B}\cdot\vec{A} ]

• Coordinate-system warning (important for torque/angular momentum later):

  • Use a right-handed coordinate system such that: [ \hat{x}\times\hat{y}=\hat{z} ]

  • If instead (\hat{x}\times\hat{y}=-\hat{z}), later directions will be wrong.

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6) 3D motion decomposed into independent 1D motions

• Position: [ \vec{r}(t) = x(t)\hat{x} + y(t)\hat{y} + z(t)\hat{z} ]

• Velocity: [ \vec{v}(t)=\frac{d\vec{r}}{dt}=\dot{x}(t)\hat{x}+\dot{y}(t)\hat{y}+\dot{z}(t)\hat{z} ]

• Acceleration: [ \vec{a}(t)=\frac{d\vec{v}}{dt}=\ddot{x}(t)\hat{x}+\ddot{y}(t)\hat{y}+\ddot{z}(t)\hat{z} ]

• Core lesson:

  • A complex 3D trajectory can be analyzed as three independent 1D motions along perpendicular axes.
  • For projectile motion (in a plane), this reduces to 2D (x and y).

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7) Projectile motion worked by x-y decomposition (2D case)

• Setup:
  • Throw with initial speed (v_0) at angle (\alpha).
  • Components:
    • (v_{0x} = v_0\cos\alpha)
    • (v_{0y} = v_0\sin\alpha)
• Choose axes:
  • x-axis horizontal
  • y-axis vertical upward (lecture takes gravity as negative in y)

x-direction (no acceleration)

• Initial conditions (chosen):
  • (x(0)=0)
  • (v_x(0)=v_0\cos\alpha)
  • (a_x=0)

• Results:

  • Position: [ x(t)=v_0\cos\alpha\; t ]

  • Velocity: [ v_x(t)=v_0\cos\alpha \quad (\text{constant}) ]

  • Acceleration: [ a_x(t)=0 ]

y-direction (gravity acts)

• Initial conditions (chosen):
  • (y(0)=0)
  • (v_y(0)=v_0\sin\alpha)
  • Gravity acceleration:
    • (a_y = -g) (with (g=+9.80) by the lecture’s sign convention)

• Results:

  • Position: [ y(t)=v_0\sin\alpha\; t-\frac{1}{2}gt^2 ]

  • Velocity: [ v_y(t)=v_0\sin\alpha - gt ]

  • Acceleration: [ a_y(t)=-g ]

• Conceptual consequences (ideal conditions):

  • In the absence of air drag/friction, (v_x) is constant for the entire flight.
  • x-motion is independent of what happens in y.
  • Therefore, matching horizontal velocity lets you “catch” the projectile at the same horizontal position when y returns to the ground level.

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8) Demonstration experiment (visual proof of x-y independence)

• Setup described:
  • A device/gun shoots a golf ball upward, tuned so that with the correct timing/angle it lands back at a specific point (returns to the same location).
• Then:
  • A cart is pushed; a switch triggers the golf gun when the cart reaches a certain point.
  • The ball is launched “straight up” relative to the cart, while the cart provides the same horizontal velocity as the ball.
• Principle being demonstrated:
  • Because x-motion is uniform and independent, the ball stays “under” the cart horizontally and lands back into it (like hands catching a thrown ball in motion).

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Speakers / sources featured

• Walter Lewin (MIT lecture speaker; primary narrator)
• “Class” / students appear as an audience response source (e.g., “Zero.”)