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Vector Components
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Addition of multiple vectors via the graphical solution is complex
B E It’s easier to use the mathematical method that will be described now
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Mathematical Method All vectors are placed on the origin of a Cartesian Coordinates System. Each Vector is replaced with it components X Y B C D A
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The vector components of vector A are two perpendicular vectors Ax and Ay that are parallel to X and Y axis respectively, add their resultant vector is equal to A. y x
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Calculation of vector components
Module Pythagoras Note that the angle is not always marked relative to positive X axis direction Y θ X
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Calculate vector A components
Example Calculate vector A components x y
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Vector representation depend on the chosen coordinate system (CS)
Vector representation depend on the chosen coordinate system (CS). Different CS does not change the vector, but only it’s representation x y x’ y’ ax ay ax’ ay’ The vector magnitude is independent of the CS, hence we have freedom to select the CS to ease the calculation
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Vector addition, is done by adding the respective components
β θ X Sum of the respective components yield the resultant vector components, hence we can find the resultant vector
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Y β θ X
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The unit vector is marked with a ^ sign above the letter
לסימן קוראים בשם "כובע" או "גג"
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It is described by it’s components as any vector.
Unit vector is a vector the has a magnitude of 1 with no units, and has a direction It is described by it’s components as any vector.
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Representing a vector by it’s components is as follows:
The CS unit vectors are unit vectors pointing parallel to the X,Y and Z axes: x, y, z z y x Representing a vector by it’s components is as follows:
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Determine unit vector components of A
Draw the vector on the CS z Determine a vector which is the same direction of vector A but has a magnitude of 1 y x
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Magnitude of A z y x
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Divide each component by the vector magnitude, results in a unit vector having same direction a vector A x y z
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Check your understanding
x y 3 cm A B 6 cm 450 Calculate: The components of vectors A and B The resultant vector components The resultant vector magnitude and direction Find the unit vector parallel to the resultant vector A vector with magnitude 10 that is parallel to the resultant vector
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The components of vectors A and B y B 6 cm
450 x 3 cm
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The resultant vector components y B 6 cm
450 x 3 cm
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The resultant vector magnitude and direction y
4.37 4.2 740 x -1.2 Above Negative X Axis
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Find the unit vector parallel to the resultant vector y
4.37 4.2 740 x -1.2
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A vector with magnitude 10 that is parallel to the resultant vector
Unit vector parallel to resultant vector
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The End!
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