Vector Addition by Components — Comprehensive Biomechanics Notes

Scalars vs vectors

• Scalars have magnitude only; they may be negative as numbers, but they do not have direction in space. Examples: work, energy, speed (scalar), etc.

• Vectors have both magnitude and direction; they are essential in biomechanics to account for both size and direction of quantities like force, velocity, and torque.

• Negative quantities can appear as scalars (e.g., negative work indicating energy decrease) and as components of vectors (negative x or y components indicating direction relative to the coordinate axes).

• Distinguish magnitude vs scalar: magnitudes are the positive sizes of a quantity; scalars can be negative and are used as part of vector expressions (e.g., work negative meaning energy decrease).

• Example: work is a scalar quantity defined as the change in energy, and negative work implies energy has decreased: $W = riangle E$ with negative W indicating a loss in energy.

Vector representations and conventions

• Two primary vector expression forms:

  • Magnitude and direction (angle of projection).

  • Components (x, y, z components in a coordinate system).

• Converting between forms is standard; component form is often preferred for addition because it scales cleanly to any number of vectors.

• Angle of projection conventions:

  • Angles are typically measured from the positive x-axis.

  • Clockwise rotation is treated as negative; counterclockwise as positive.

  • When dealing with torque, clockwise torque is usually negative and counterclockwise is positive.

  • The coordinate system setup governs the sign of components and resulting angle.

• In 3D, include a z component; in this course the focus is on the x-y plane for vector addition.

Express vectors in components (example setup)

• Given two coplanar forces, express each in x and y components:

  • For a vector with magnitude $F$ and projection angle $ heta$ (from +x axis):

  • $F_x = F \, \cos\theta$

  • $F_y = F \, \sin\theta$

• Example components (from the lecture):

  • Jetpack force: magnitude $F_1 = 2500$ N at angle $\theta = 65^{\circ}$

  • Anvil force: magnitude $F_2 = 1800$ N directed straight downward (negative y)

• Compute components:

  • $F<em>{1x} = 2500 \cos 65^{\circ}, \quad F</em>{1y} = 2500 \sin 65^{\circ}$

  • $F<em>{2x} = 0, \quad F</em>{2y} = -1800$

• In numerical terms (approximate):

  • $F_{1x} \approx 2500 \times 0.4226 \approx 1{,}057\ \text{N}$

  • $F_{1y} \approx 2500 \times 0.9063 \approx 2{,}266\ \text{N}$

  • $F<em>{2x} = 0, \quad F</em>{2y} = -1800\ \text{N}$

Net force in components and magnitude

• Net force components (sum of x’s and y’s):

  • $F<em>{\text{net},x} = F</em>{1x} + F_{2x} = 1{,}057 + 0 \approx 1{,}057\ \text{N}$

  • $F<em>{\text{net},y} = F</em>{1y} + F_{2y} = 2{,}266 - 1{,}800 \approx 466\ \text{N}$

• Net force in component form:

  • $\mathbf{F}<em>{\text{net}} = \langle F</em>{\text{net},x}, F_{\text{net},y} \rangle \approx \langle 1057, \ 466 \rangle \ \text{N}$

Magnitude and angle of the net force

• Magnitude of the net force:

  • $|\mathbf{F}<em>{\text{net}}| = \sqrt{F</em>{\text{net},x}^2 + F_{\text{net},y}^2} <br>= \sqrt{1057^2 + 466^2} \approx 1.15 \times 10^3 \text{ N}$

• Angle of projection (relative to +x axis):

  • $\theta = \tan^{-1}\left(\frac{F<em>{\text{net},y}}{F</em>{\text{net},x}}\right) = \tan^{-1}\left(\frac{466}{1057}\right) \approx 23.8^{\circ}$

• Note on quadrant: since both components are positive, the vector lies in Quadrant I (positive x and positive y).

Graphical methods for vector addition

• Tip-to-tail method (graphical, quantitative):

  • Arrange vectors end-to-end and draw from the start of the first to the end of the last to obtain the resultant.

  • Works for any number of vectors; order (the sequence) does not affect the final resultant for vector addition.

• Parallelogram method (for two vectors):

  • Place both vectors tail-to-tail and construct the parallelogram; the diagonal gives the resultant.

• Cross-product note (not for addition):

  • The order of operands matters in cross products: for torque calculations, use (\mathbf{r} \times \mathbf{F}) rather than (\mathbf{F} \times \mathbf{r}).

• Law of cosines as an alternative approach (less general, sometimes convenient):

  • For two vectors a and b with angle gamma between them (the included angle):

  • $c^2 = a^2 + b^2 - 2ab \cos \gamma$

  • Here, c is the magnitude of the resultant vector when the two are arranged head-to-tail in a triangle.

  • Parallelogram approach variant uses the angle theta between them tail-to-tail and yields:

  • $c^2 = a^2 + b^2 + 2ab \cos \theta$

  • Relationship between gamma and theta: (\gamma = 180^{\circ} - \theta).

• Matrix method (alternative quantitative approach):

  • Build a matrix of the components and perform matrix operations to sum vectors; equivalent to summing components directly.

• Takeaway: for most biomechanics problems, addition by components (and using the Pythagorean and arctangent relationships) is the most general and scalable approach.

Applications to biomechanics and torque generation

• Muscular contributions: muscles can pull in different directions; the resultant net force on a joint depends on the vector sum of individual muscle forces.

• Example: pectoralis major has fibers with different directions; activation levels combine to produce a net tendon pull and torque.

• Changing joint posture (e.g., bench press incline/decline) changes the plane of motion and the relative contributions of different muscle fiber directions, altering the net torque produced.

• Key idea: break forces into perpendicular components relative to the lever arm; decompose into perpendicular (torque-producing) and parallel (compressive or joint-closing) components.

• Resultant force analysis helps determine which postures maximize desired torque while managing compressive forces on joints.

Coplanar vectors and the knee pulley analogy (biomechanics example)

• Visualizing a knee sagittal view: the quadriceps tendon pulls on the patella, redirecting force across the knee like a pulley.

• Consequences of the redirection: a compressive force pushing the patella into the femur’s articular cartilage, which is relevant for osteoarthritis and joint pain.

• Practical exercise setup (from the lecture): assume the quadriceps can maintain a constant 300 N force through the range of motion.

• Question posed: as knee flexion increases (more deep into flexion), how does the net force from the two red vectors change (remain the same, increase, or decrease)?

  • Students are asked to sketch the vector composition for differing knee angles and discuss how the resultant vector changes.

• Real-world relevance: exercise prescription could mitigate compressive forces by selecting joint angles that reduce net anterior-posterior loading on the patellofemoral joint.

Practical notes and common pitfalls

• Degree vs radian mode on calculators:

  • In this course, work in degree mode for angles unless radian-mode is specifically required later.

  • Using radians by mistake can yield incorrect results when applying trigonometric functions with degree inputs.

• Degree reference and angle signs:

  • Always reference angles to the +x axis; clockwise is negative, counterclockwise is positive.

• Quadrants and inverse trig:

  • When using inverse tangent, be mindful of the quadrant of the resulting vector; use quadrant checks or the atan2 function if available.

• Sign conventions for torques:

  • Clockwise torques are typically negative; counterclockwise torques are positive; be consistent with the chosen coordinate system.

• Summary of the main workflow for two coplanar vectors:

  • Express each vector in components: $F<em>{ix} = F</em>i \cos\theta<em>i, \quad F</em>{iy} = F<em>i \sin\theta</em>i$

  • Sum components: $F<em>{\text{net},x} = \sum F</em>{ix}, \quad F<em>{\text{net},y} = \sum F</em>{iy}$

  • Compute magnitude and angle: $|\mathbf{F}<em>{\text{net}}| = \sqrt{F</em>{\text{net},x}^2 + F<em>{\text{net},y}^2}, \quad \theta = \tan^{-1}\left(\frac{F</em>{\text{net},y}}{F_{\text{net},x}}\right)$

• Open exercise reminder (friday topic): analyze how net force changes with knee flexion given a constant quadriceps force and two red force vectors; sketch and discuss.

Quick reference formulas (biomechanics vector sums)

• Component form of a vector: $\mathbf{F} = \langle F<em>x, F</em>y \rangle,\quad F<em>x = F \cos\theta,\quad F</em>y = F \sin\theta$

• Net components: $F<em>{\text{net},x} = \sum</em>i F<em>{ix}, \quad F</em>{\text{net},y} = \sum<em>i F</em>{iy}$

• Net magnitude: $|\mathbf{F}<em>{\text{net}}| = \sqrt{F</em>{\text{net},x}^2 + F_{\text{net},y}^2}$

• Net angle: $\theta = \tan^{-1}\left(\frac{F<em>{\text{net},y}}{F</em>{\text{net},x}}\right)$

• Law of cosines (two vectors, triangle method):$c^2 = a^2 + b^2 - 2ab \cos\gamma\,,$ where gamma is the angle between the two vectors.

• Law of cosines (parallelogram tail-to-tail, theta):$c^2 = a^2 + b^2 + 2ab \cos\theta\,,$ with theta the angle between the vectors when arranged tail-to-tail; note that (\gamma = 180^{\circ} - \theta).

• Cross product note (torque): $\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}\,,$ order matters for the cross product (but not for simple vector addition).

End-of-lecture prompts

• How does changing the knee angle affect the net force on the patellofemoral joint, given a constant quadriceps force? Sketch and explain using vector addition concepts.

• Consider how aligning muscle fiber directions with the plane of motion affects the torque and potential joint loading in exercises like bench press variations.