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Components of Vectors Section 4.2

Components of Vectors Section 4.2. Physics. Components of Vectors. Breaking vectors into their component parts: Vector Resolution The magnitude and sign of component vectors: Components Components are found using trigonometry. Component Vectors. y. A. A y. Θ. A x. x.

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Components of Vectors Section 4.2

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  1. Components of VectorsSection 4.2 Physics

  2. Components of Vectors • Breaking vectors into their component parts: Vector Resolution • The magnitude and sign of component vectors: Components • Components are found using trigonometry.

  3. Component Vectors y A Ay Θ Ax x

  4. Component Vectors • Ay = A sin Θ • Sin = Opposite / Hypotenuse (SOH) • Ax = A cos Θ • Cos = Adjacent / Hypotenuse (CAH)

  5. Vector Quadrants • The sign of a component depends upon which of the four quadrants the component is in. • When the angle that a vector makes with the x-axis is larger than 90° -- the sign of one or more components is negative.

  6. First Quadrant Second Quadrant Ax < 0 Ay > 0 Ax > 0 Ay > 0 Both Components are positive x Fourth Quadrant Third Quadrant Ax < 0 Ay < 0 Ax > 0 Ay < 0 y

  7. Example Problem • A bus travels 23.0 km on a straight road that is 30° north of east. What are the east and north components of its displacement? N A Ay W Θ Ax E S

  8. Problems • What are the components of a vector of magnitude 1.5 m at an angle of 35° from the positive x-axis? • A hiker walks 14.7 km at an angle 35° south of east. Find the east and north components of this walk.

  9. More Problems • An airplane flies at 65 m/s in the direction 149° counterclockwise from east. What are the east and north components of the plane’s velocity? • A golf ball, hit from the tee, travels 325 m in a direction 25° south of the east axis. What are the east and north components of its displacement?

  10. Not All Things Travel in a Straight Line! • Algebraic Addition of Vectors: • Two or more vectors can be divided into their x and y components and the resultant vector can be found.

  11. Resultant Vector y C Cy B Resultant Vector R² = Rx² + Ry² By Ry = Ay + By + Cy Rx = Ax + Bx + Cx Ay A Ax Bx Cx x

  12. Resultant Angle y tan Θ = Rx² ÷ Ry² tan = TOA R Ry Θ = ? Rx x

  13. Problems • A powerboat heads due northwest at 13 m/s with respect to the water across a river that flows due north at 5 m/s. What is the velocity (both magnitude and direction) of the motorboat with respect to the shore?

  14. Problems • An airplane flies due south at 175 km/h with respect to the air. There is a wind blowing at 85 km/h to the east relative to the ground. What are the plane’s speed and direction with respect to the ground?

  15. Problems • An airplane flies due north at 235 km/h with respect to the air. There is a wind blowing at 65 km/h to the northeast with respect to the ground. What are the plane’s speed and direction with respect to the ground?

  16. The End

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