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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:

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Chapter 3 vector l.jpg
Chapter 3. Vector

1. Adding Vectors Geometrically      

2. Components of Vectors      

3. Unit Vectors      

4. Adding Vectors by Components        

5. Multiplying Vectors 


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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 .



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Two important properties of vector additions

(1) Commutative law:

(2) Associative law:



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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?


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Unit Vectors

The unit vectors are dimensionless vectors that point in the direction along a coordinate axis that is chosen to be positive


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How to describe a two-dimension vector?

Vector Components:The projection of a vector on an axis is called its component .


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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


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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.




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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?


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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.





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Example

What is the angle between

and ?


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The Vector Product (cross product )

(3) Direction is

determined by right-hand rule

(1) Cross production is a vector

(2) Magnitude is


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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.


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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                 ?


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Sample Problem

If                      and                , what is                         ?


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