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CE Statics Lecture 4

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CARTESIAN VECTORS We knew how to represent vectors in the form of Cartesian vectors in two dimensions. In this section, we will look into how vectors can be represented in Cartesian vectors form and in three dimensions. x y F FxFx FyFy i j F = Fx i + Fy j

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Coordinate System In this course, the right-handed coordinate system will be used. Thumb is directed towards the +ve z-axis

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Rectangular Components of a Vector If F is a vector, then it can be resolved into x, y, and z components depending on the direction of F with respect to x, y, and z axes. F = Fz + F F = Fx + Fy then, F = Fx + Fy + Fz (the three components of F) F F FxFx FyFy FzFz x y z

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Unit Vector has a magnitude of 1. If F is a vector having a magnitude of f 0, then the unit vector which has the same direction as F is represented by: u F = (F / f) then, F = f u F f defines the magnitude of vector F u F is dimensionless since F and f have the same set of units. F f uFuF

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Cartesian Unit Vector The Cartesian unit vectors i, j, and k are used to designate the directions of x, y, and z axes, respectively. The sense of the unit vectors will be described by minus (-) or plus (+) signs depending on whether the vectors are pointing along the +ve or –ve x, y, and z axes. i j k x y z

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Cartesian Vector Representation The three components of F can be represented by: F = Fx i + Fy j + Fz k F f F xi F yj F zk x y z fxfx fyfy fzfz i j k

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Magnitude of a Cartesian Vector f = f 2 + fz 2 f = fx 2 + fy 2 then, f = fx 2 + fy 2 + fz 2 Hence, the magnitude of vector F is the square root of the sum of the squares of its components.

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Direction of a Cartesian Vector The orientation of vector F is determined by angles , , and measured between the tail of vector F and the +ve x, y, and z axes, respectively. Cos ( ) = fx / f Cos ( ) = fy / f Cos ( ) = fz / f , , and are between 0 and 180 F f xi f yj f zk x y z

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Another way of determining the direction of F is by forming a unit vector in the direction of F. If, F = fx i +fy j + fz k u F = F / f = (fx/f) i + (fy/f) j + (fz/f) k where, f = fx 2 + fy 2 + fz 2 then, u F = cos ( ) i + cos ( ) j + cos ( ) k since f = fx 2 + fy 2 + fz 2 and u F =1 then, cos 2 ( ) + cos 2 ( ) + cos 2 ( ) = 1 if the magnitude and direction of F are known, then F = f u F = f cos ( ) i + f cos ( ) j + f cos ( ) k = fx i +fy j + fz k

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Addition and Subtraction of Cartesian Vectors If, F = Fx i + Fy j + Fz k H = Hx i + Hy j + Hz k then, F + H = R = (Fx + Hx) i + (Fy + Hy) j + (Fz + Hz) k F - H = R = (Fx - Hx) i + (Fy - Hy) j + (Fz - Hz) k F (f x + h x )(f y + h y ) (f z + h z ) x y z H R

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Concurrent Force Systems In general, F R = F x i + F y j + F z k

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Examples Example 2.8 Example 2.9 Example 2.10 Example 2.11 Problem 2.62 Problem 2.69

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