Vector Components

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

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

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

Calculation of vector components Module Pythagoras Note that the angle is not always marked relative to positive X axis direction Y θ X

Calculate vector A components Example Calculate vector A components x y

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

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

Y β θ X

The unit vector is marked with a ^ sign above the letter לסימן קוראים בשם "כובע" או "גג"

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.

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:

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

Magnitude of A z y x

Divide each component by the vector magnitude, results in a unit vector having same direction a vector A x y z

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

The components of vectors A and B y B 6 cm 450 x 3 cm

The resultant vector components y B 6 cm 450 x 3 cm

The resultant vector magnitude and direction y 4.37 4.2 740 x -1.2 Above Negative X Axis

Find the unit vector parallel to the resultant vector y 4.37 4.2 740 x -1.2

A vector with magnitude 10 that is parallel to the resultant vector Unit vector parallel to resultant vector

The End!