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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 = 6.0 m x 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. Fy Fy = (2.0 N) sin(60°) = 1.7 N 60° Fx • x = r cos • y = r sin Fx = (2.0 N) cos(60°) = 1.0 N This is called projection onto the axes. 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. A ( Ax , Ay ) A (4.23, 3.66) v (v x , v y , v z ) v (2.1, 10.5, 3.2) Component Addition A vector equation is actually a set of equations. • One equation for each component • Components can be added like the vectors themselves A (3 N,0 N ) ( Ax , Ay ) B ( 2 N , 4 N ) ( Bx , B y ) C A B C x Ax Bx 5 N C y Ay B y 4 N C (C x , C y ) (5 N,4 N) 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. d d x2 d y2 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 27 4.1 N tan Ay Ax Components to Angles Find the magnitude and angle of a vector with components x = -5.0 N, y = 3.3 N. L x y 2 2 y x = -5.0 N 2 L x y 2 y = 3.3 N 2 tan y / x tan 1 ( y / x) L L = 6.0 N = 33o above the negative x-axis x Alternate Axes Projection works on other choices for the coordinate axes. Other axes may make more sense for a particular physics problem. y’ f x’ f next