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**Adding Vectors by Components**

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Dimension The number of coordinates needed to specify a point (Ex)

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**One dimension = position on a line**

Two dimensions = position on a plane Three dimensions = position in 3D-space Four dimensions = 3D + time

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**To describe a direction**

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**x degrees north of east = Start at east and rotate x degrees toward north.**

x degrees east of north = Start at north and rotate x degrees toward east.

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**Revisit Vectors 1) Vectors in one dimension Fnet = ? F1 = 120 N**

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**2) Vectors in two dimensions**

Fnet = ?

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Component Vectors Visualizing Vectors in 2 Dimensions | Mechanics | Khan Academy Vector resolution Breaking a vector into the x- and y- components Rx = Ry = vector R = Rx = the x component of a vector, R Ry = the y component of a vector, R

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Example Find the x- and y-component of R if R = 15 cm and ө = 50˚.

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**Adding Vectors by Components**

Add the vectors. 15 m/s 13 m/s 20˚ 40˚

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y-axis 13 m/s resultant vector 15 m/s 20˚ 40˚ x-axis

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y-axis 20˚ 40˚ x-axis

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y-axis 20˚ 40˚ x-axis

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**y-axis 13 m/s 20˚ 15 m/s 40˚ x-axis 13sin 20˚ 13cos20˚ 15sin 40˚**

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y-axis 13sin 20˚ 20˚ 15sin 40˚ 40˚ x-axis 15cos40˚ 13cos20˚

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y-axis 13sin 20˚ resultant vector 15sin 40˚ x-axis 15cos40˚ 13cos20˚

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Example The table below represents a set of force vectors. These vectors begin at the origin of a coordinate system, and end at the coordinates given in the table. What is the magnitude of the resultant of the sum of these three vectors? What is the size of the angle, ө, that the resultant makes with the horizontal (x-axis)? Vector # x-value (N) y-value (N) A 0.0 6.0 B 5.0 C -10.0

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**Find the resultant vector.**

B(-8, -3) C(9, -10)

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Answer 8.5 R (6, -6)

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**Find the resultant vector.**

55° 40° 50° 4N 5N B C

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**25° vectors x component (N) y component (N) A 3·cos 55 = 1.7**

3·sin 55 = 2.5 B 4·cos 220 = -3.1 4·sin 220 = -2.6 C 5·cos 310 = 3.2 5·sin 310 = -3.8 Resultant R = 1.8 = -3.9 25° 4.3N R (1.8, -3.9)

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Example A GPS receiver indicates that your home is 15.0 km and 40.0˚ north of west, but the only path through the woods leads directly north. If you follow the path 5.0 km before it opens into a field, how far, and in what direction, would you have to walk to reach your home? *north of west = north from west

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**Ax = 0 Bx Rx = 15 cos 140˚ = - 11.5 Ay = 5 By Ry = 15 sin 140˚ = 9.6**

x km, B ө ˚ R, 15 km 5 km, A 40˚ Ax = 0 Bx Rx = 15 cos 140˚ = Ay = 5 By Ry = 15 sin 140˚ = 9.6 Ax + Bx = Rx 0 + Bx = Bx = Ay + By = Ry 5 + By = 9.6 By = 4.6

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**Ay + By = Ry Ax + Bx = Rx 5 + By = 9.6 0 + Bx = - 11.5 By = 4.6**

vectors x component (km) y component (km) A 5 B Bx By R 15 cos 140˚ = 15 sin 140˚ = 9.6 x km, B ө ˚ 4.6 km km Ay + By = Ry 5 + By = 9.6 By = 4.6 Ax + Bx = Rx 0 + Bx = Bx = x = ө + 90˚ = cos-1 (-11.5/12.4) =158˚ ө = 158˚ - 90˚ = 68˚

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Example A football player runs directly down the field for 35 m before turning an additonal 15 m before getting tackled. What is the magnitude and direction of the runner’s total displacement?

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