Kinematics 1B: Vectors in 3D and Component Decomposition

Introduction

• Welcome to physics lecture 101.
• Lecturer: John Siakobelos.
• I will be your note-taker for this course.

Unit Focus: Kinematics 1B

• First unit: Kinematics 1B.
• Key topics: vectors in three dimensions (3D).
• Next step: learn how to split any quantity into x and y components (and extend to z in 3D) to analyze motion.
• Goal mentioned: use component addition to determine final and initial velocity.

Vectors in Three Dimensions (3D)

• A vector in 3D has components along three perpendicular axes: x, y, and z.
• Notation often used:
  • Position vector: $
    vec{r} = \langle x, y, z \rangle$
  • Velocity vector: $
    vec{v} = \langle vx, vy, v_z \rangle$
• Vectors convey both magnitude and direction, and component form allows easy arithmetic.

Decomposition into Components

• Core idea: any vector can be expressed as a sum of its axis-aligned components.
• In 3D, components along x, y, z allow reconstruction:
  • \nvec{v} = \langle vx, vy, v_z \rangle
• In 2D (x and y), decomposition uses two components:
  • $v<em>x = v \cos\theta, \quad v</em>y = v \sin\theta$
  • Reconstruct: \nvec{v} = \langle vx, vy \rangle
• In 3D, you may also describe direction with angles \alpha, \beta, \gamma (direction cosines) relative to the x, y, z axes:
  • $v<em>x = v \cos\alpha, \quad v</em>y = v \cos\beta, \quad v_z = v \cos\gamma$
• Practical approach: project any motion quantity onto the coordinate axes to analyze each component separately, then combine via vector addition.

Adding Components and Velocity Calculations

• Vector addition is performed component-wise:
  • If \nvec{a} = \langle ax, ay, az \rangle, \; \nvec{b} = \langle bx, by, bz \rangle, then
  • \nvec{a} + \nvec{b} = \langle ax + bx, ay + by, az + bz \rangle
• Initial and final velocity vectors relate by their difference:
  • \Delta \nvec{v} = \nvec{v}{\text{final}} - \nvec{v}{\text{initial}}
• Magnitude of a 3D velocity vector:
  • |\nvec{v}| = \sqrt{vx^2 + vy^2 + v_z^2}
• When only horizontal components are considered (2D), the magnitude becomes:
  • |\nvec{v}{2D}| = \sqrt{vx^2 + v_y^2}

Examples and Scenarios (conceptual)

• Example 1 (2D): A velocity vector with magnitude $v$ at angle $\theta$ from the x-axis has components:
  • $v<em>x = v \cos\theta, \quad v</em>y = v \sin\theta$
• Example 2 (3D): A velocity vector with magnitude $v$ making direction cosines $(\alpha, \beta, \gamma)$ with axes has:
  • $v<em>x = v \cos\alpha, \quad v</em>y = v \cos\beta, \quad v_z = v \cos\gamma$
• Example 3 (combining motions): If two velocity vectors are known as \nvec{u} = \langle ux, uy, uz \rangle and \nvec{v} = \langle vx, vy, vz \rangle, then the resultant is \nvec{r} = \nvec{u} + \nvec{v} = \langle ux + vx,\; uy + vy,\; uz + vz \rangle
• Caveat: ensure consistent units across components when adding or subtracting vectors.

Notation and Key Formulas

essential formulas to remember:

• Vector components in 3D: \nvec{v} = \langle vx, vy, v_z \rangle
• Magnitude of a 3D vector: |\nvec{v}| = \sqrt{vx^2 + vy^2 + v_z^2}
• Vector addition (component-wise): \nvec{a} + \nvec{b} = \langle ax + bx, ay + by, az + bz \rangle
• Initial to final velocity change: \Delta \nvec{v} = \nvec{v}{\text{final}} - \nvec{v}{\text{initial}}
• 2D component relations (for a vector of magnitude $v$ at angle $\theta$): $v<em>x = v \cos\theta, \quad v</em>y = v \sin\theta$
• 3D direction cosines (angles with axes): $v<em>x = v \cos\alpha, \quad v</em>y = v \cos\beta, \quad v_z = v \cos\gamma$
• Projection concept: components are projections of the vector onto coordinate axes.

Connections to Prior Knowledge and Real-World Relevance

• Builds on basic geometry: understanding magnitude and direction of vectors.
• Essential for analyzing motion in physics, engineering, computer graphics, aerospace, sports science, and robotics.
• Real-world relevance: predicting trajectories, determining resultant motion from multiple velocity sources, and decomposing forces in equilibrium problems.

Practical Implications and Measurement Considerations

• Precision in decomposing vectors matters: small errors in components can lead to large errors in resultant quantities due to squaring in magnitude calculations.
• In experiments, always carry units consistently and account for measurement uncertainty in each component.
• When plotting trajectories or simulating motion, 3D vector decomposition enables accurate modeling of direction changes and resultant speeds.

Summary of Key Takeaways

• Kinematics 1B focuses on vectors in 3D and decomposing quantities into x, y (and z) components.
• Component addition allows reconstruction of original vectors and calculation of changes in velocity.
• Mastery of these concepts enables solving complex motion problems by reducing them to simpler axis-aligned analyses.

Ethical and Philosophical Considerations (educational practice)

• Emphasizes careful data handling and clear notation to avoid misinterpretation in physics work.
• Encourages rigorous, transparent reasoning when projecting vectors and interpreting results in real-world contexts.

Quick Reference: Typical Notation

• Vector: \nvec{v} = \langle vx, vy, v_z \rangle
• Magnitude: |\nvec{v}| = \sqrt{vx^2 + vy^2 + v_z^2}
• Sum: \nvec{a} + \nvec{b} = \langle ax + bx, ay + by, az + bz \rangle
• Initial to final velocity change: \Delta \nvec{v} = \nvec{v}{\text{final}} - \nvec{v}{\text{initial}}
• 2D components: $v<em>x = v \cos\theta, \; v</em>y = v \sin\theta$
• 3D direction cosines: $v<em>x = v \cos\alpha, \; v</em>y = v \cos\beta, \; v_z = v \cos\gamma$n