Vectors in Biomechanical Analysis

1. Vectors in Biomechanical Analysis Vectors represent dynamic action Vectors have two defining characteristics: Magnitude – The size or value Direction 65 MPH is speed (scalar.) Also: mass, volume, density 65 MPH due east is a vector Velocity, force, acceleration, momentum Waco aTm Houston

2. Parts of a Vector Like a comet or an arrow Tail – The beginning (non-arrowhead) of the vector. Tip – The arrowhead end of the vector.

3. Vector Representation • Vectors are represented by an arrow • The length of the arrow reflects the magnitude • The angular position of the arrow represents the vector’s direction magnitude Direction (q )

4. magnitude Direction • Magnitude is a physical quantity (20 ft/sec, 50N, 10 m/sec2, etc.) • Direction is often expressed as an angle relative to a known reference (450 above horizontal)

5. Working With Multiple Vectors • The relative size of vector quantities is shown by the relative size of the arrows representing them. 20 m/s 40 m/s The vector representing 40 m/s would be twice as long as the vector representing 20 m/s.

6. Vector Operations • Vector Composition • The process of determining a single vector from 2 or more vectors • Vector Resolution • An operation that replaces a single vector with 2 perpendicular vectors

7. Vector Operations B • Vector Composition • Tip to tail method • Linear • A. Addition (ex. impact forces) A A B = + net + = 100 N 200 N 300 N + = -200 N 100 N -100 N

8. Vector Operations • Vector Composition • Linear • A. subtraction (wind, air movements - drag) - = 10 m/sec -2 m/sec 12 m/sec Sprinting into a headwind - = 2 m/sec 8 m/sec 10 m/sec Sprinting with a tailwind

9. Vector Operations • Vector Composition • Planar • A. Right angles A --> net B B A Solve for net using Pythagorean Theorem

10. Vector Operations • Vector Composition • Planar • A. Obtuse non-right angles Q = 180° - f 1 net f --> 2 q 2 1 f Solve using the law of cosines or law of sines

11. 1350 Vector Composition Of Non Co-linear Vectors 2 tacklers are pulling on Dante Hall Player A uses a force of 200 N and. Player B is a force of 100 N acting at an angle of 1350 from vector A. Calculate the resultant of these two vectors. Vector A (vA) Vector B (vB)

12. 1350 Vector B (vB) Vector A (vA) Vector Composition Of Non Co-linear Vectors • (1) tip to tail method f q

13. 1350 net Vector B (vB) vC Vector A (vA) Vector Composition • (2) a vector drawn from tail of the first vector to the tip of the last is the resultant (net). Call the net Vector C (vC). q

14. Vector B (vB) Resultant 1350 vC q Vector A (vA) C= √(200N)2+(100N)2-2(100N)(200N)cos 450= 147.4 N Vector Composition - Non Co-linear Vectors • (3) Use the Law of Cosines to calculate the magnitude of net vector. A = vA, B = vB, and C is the magnitude of the resultant vC. q is the angle opposite vC. q= 1800 – 1350 = 450 C2= A2+ B2-(2* A* B*cos q) vC= √A2+ B2-(2* A* B*cos q)

15. Vector B (vB) Resultant b 1350 vC q Vector A (vA) Vector Composition Of Non Co-linear Vectors • (4) Since a vector has magnitude and direction, use the Law of Sines to find the direction. b is the angle opposite vB. - the angle that the resultant makes relative to vector A

16. Vector B (vB) Resultant b 1350 vC q Vector A (vA) Vector Composition Of Non Co-linear Vectors law of sines: B/sin b = C/sin q sin b = (B* sin q)/C = (100 N * sin 450)/147.4 N) sin b = 0.480 Now find ß by using the inverse sine (sin-1)

17. Vector Composition Of Non Co-linear Vectors • (5) inverse sine: expressed as sin-1 to find the angle. sin-1 (sin q ) = q • EXAMPLE: sin 300 = .5 sin-1 .5 = 300 Then: If sin b = 0.480 , b = sin-1 (.480) = 28.670 Therefore, vc 147.4 N at an angle 28.670 above Player A.