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Vectors

Vectors. An Introduction. There are two kinds of quantities…. Vectors are quantities that have both magnitude and direction (e.g., displacement, velocity, acceleration). Scalars are quantities that have magnitude only (e.g., position, speed, time, mass). →. Vector: R R. R.

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Vectors

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  1. Vectors An Introduction

  2. There are two kinds of quantities… • Vectors are quantities that have both magnitude and direction (e.g., displacement, velocity, acceleration). • Scalars are quantities that have magnitude only (e.g., position, speed, time, mass).

  3. → • Vector:R R R head tail Notating vectors This is how you draw a vector.

  4. Notating scalars • Scalar:R There is no standard way to draw a scalar!

  5. θ x θ x B A Direction of Vectors

  6. y II 90o < θ < 180o I 0 < θ < 90o x θ θ θ θ III 180o < θ < 270o IV 270o < θ < 360o Vector angle ranges

  7. Magnitude of Vectors • The best way to describe the magnitude of a vector is to measure the length of the vector. • The length of the vector is proportional to the magnitude of the quantity it represents.

  8. If vector A represents a displacement of three miles to the north… A B Then vector B, which is twice as long, would represent a displacement of six miles to the north! Magnitude of Vectors

  9. Equal Vectors Equal vectors have the same length and direction, and represent the same quantity (such as force or velocity).

  10. A -A Inverse Vectors Inverse vectors have the same length, but opposite direction.

  11. A θ x Ax Vectors: x-component Ax = A cos θ

  12. A θ Ay x Vectors: y-component Ay = A sin θ

  13. Vectors: angle θ = tan-1 (Ry/Rx) y Ry θ x Rx

  14. R = √ (Rx2 + Ry2) Vectors: magnitude y R Ry x Rx

  15. Graphical Addition of Vectors • You’ll need: Graph paper Pencils Ruler Protractor

  16. B A R Graphical Addition of Vectors A + B = R R is called the resultant vector!

  17. The Resultant and the Equilibrant The sum of two or more vectors is called the resultant vector. The resultant vector can replace the vectors from which it is derived. The resultant is completely canceled out by adding it to its inverse, which is called the equilibrant.

  18. B E A R Graphical Addition of Vectors A + B = R E is called the equilibrant vector!

  19. Component Addition of Vectors • Resolve each vector into its x- and y-components. Ax = Acosθ Ay = Asinθ Bx = Bcosθ By = Bsinθ Cx = Ccosθ Cy = Csinθ etc. • Add the x-components (Ax, Bx, etc.) together to get Rx and the y-components (Ay, By, etc.) to get Ry.

  20. Component Addition of Vectors • Calculate the magnitude of the resultant with the Pythagorean Theorem R = √(Rx2 + Ry2) • Determine the angle with the equation θ = tan-1 Ry/Rx.

  21. Vs Vw Vt = Vs +Vw Relative Motion S = swimmer W = water

  22. Vs Vw Vt = Vs +Vw Relative Motion

  23. Vs Vw Vt = Vs +Vw Relative Motion

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