Vector Components

Coordinates Vectors can be described in terms of coordinates. 6.0 km east and 3.4 km south 1 N forward, 2 N left, 2 N up Coordinates are associated with axes in a graph. y x x = 6.0 m y = -3.4 m

Use of Angles Find the components of vector of magnitude 2.0 N at 60° up from the x-axis. Use trigonometry to convert vectors into components. x = r cos  y = r sin  This is called projection onto the axes. Fy Fx Fy = (2.0 N) sin(60°) = 1.7 N 60° Fx = (2.0 N) cos(60°) = 1.0 N

Ordered Set The value of the vector in each coordinate can be grouped as a set. Each element of the set corresponds to one coordinate. 2-dimensional 3-dimensional The elements, called components, are scalars, not vectors.

Component Addition A vector equation is actually a set of equations. One equation for each component Components can be added like the vectors themselves

Vector Length Vector components can be used to determine the magnitude of a vector. The square of the length of the vector is the sum of the squares of the components. 4.6 N 2.1 N 4.1 N

Vector Direction Vector components can also be used to determine the direction of a vector. The tangent of the angle from the x-axis is the ratio of the y-component divided by the x-component. 4.6 N 2.1 N q = 27 4.1 N

Components to Angles Find the magnitude and angle of a vector with components x = -5.0 N, y = 3.3 N. y x x = -5.0 N L y = 3.3 N  L = 6.0 N  = 33o above the negative x-axis

Alternate Axes Projection works on other choices for the coordinate axes. Other axes may make more sense for a particular physics problem. y’ x’  f f next