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##### Chapter 3. Vector

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**Chapter 3. Vector**1. Adding Vectors Geometrically 2. Components of Vectors 3. Unit Vectors 4. Adding Vectors by Components 5. Multiplying Vectors**Adding Vectors Graphically**General procedure for adding two vectors graphically: • (1) On paper, sketch vector to some convenient scale and at the proper angle. • (2) Sketch vector to the same scale, with its tail at the head of vector , again at the proper angle. • (3) The vector sum is the vector that extends from the tail of to the head of . General procedure for adding two vectors graphically: • (1) On paper, sketch vector to some convenient scale and at the proper angle. • (2) Sketch vector to the same scale, with its tail at the head of vector , again at the proper angle. • (3) The vector sum is the vector that extends from the tail of to the head of . General procedure for adding two vectors graphically: • (1) On paper, sketch vector to some convenient scale and at the proper angle. • (2) Sketch vector to the same scale, with its tail at the head of vector , again at the proper angle. • (3) The vector sum is the vector that extends from the tail of to the head of . General procedure for adding two vectors graphically: • (1) On paper, sketch vector to some convenient scale and at the proper angle. • (2) Sketch vector to the same scale, with its tail at the head of vector , again at the proper angle. • (3) The vector sum is the vector that extends from the tail of to the head of . General procedure for adding two vectors graphically: • (1) On paper, sketch vector to some convenient scale and at the proper angle. • (2) Sketch vector to the same scale, with its tail at the head of vector , again at the proper angle. • (3) The vector sum is the vector that extends from the tail of to the head of .**Two important properties of vector additions**(1) Commutative law: (2) Associative law:**Check Your Understanding**Two vectors, A and B, are added by means of vector addition to give a resultant vector R: R=A+B. The magnitudes of A and B are 3 and 8 m, but they can have any orientation. What is (a) the maximum possible value for the magnitude of R? (b) the minimum possible value for the magnitude of R?**Unit Vectors**The unit vectors are dimensionless vectors that point in the direction along a coordinate axis that is chosen to be positive**How to describe a two-dimension vector?**Vector Components:The projection of a vector on an axis is called its component .**Properties of vector component**• The vector components of the vector depend on the orientation of the axes used as a reference. • A scalar is a mathematical quantity whose value does not depend on the orientation of a coordinate system. The magnitude of a vector is a true scalar since it does not change when the coordinate axis is rotated. However, the components of vector (Ax, Ay) and (Ax′, Ay′), are not scalars. • It is possible for one of the components of a vector to be zero. This does not mean that the vector itself is zero, however. For a vector to be zero, every vector component must individually be zero. • Two vectors are equal if, and only if, they have the same magnitude and direction**Example 1Finding the Components of a Vector**A displacement vectorr has a magnitude of r 175 m and points at an angle of 50.0° relative to the x axis in Figure. Find the x and y components of this vector.**Reconstructing a Vector from Components**; Magnitude: Direction:**Check Your Understanding**• Two vectors, A and B, have vector components that are shown (to the same scale) in the first row of drawings. Which vector R in the second row of drawings is the vector sum of A and B?**Example 2The Component Method of Vector Addition**A jogger runs 145 m in a direction 20.0° east of north (displacement vector A) and then 105 m in a direction 35.0° south of east (displacement vector B). Determine the magnitude and direction of the resultant vector C for these two displacements.**Example**What is the angle between and ?**The Vector Product (cross product )**(3) Direction is determined by right-hand rule (1) Cross production is a vector (2) Magnitude is**Property of vector cross product**• The order of the vector multiplication is important. If two vectors are parallel or anti-parallel, . If two vectors are perpendicular to each other , the magnitude of their cross product is maximum.**Sample Problem**In Fig. 3-22, vector lies in the xy plane, has a magnitude of 18 units and points in a direction 250° from the +x direction. Also, vector has a magnitude of 12 units and points in the +z direction. What is the vector product ?**Sample Problem**If and , what is ?