Chapter 6 Vectors
6.1 Graphical Method of Vector Addition
Scalar quantity and unit
Vector quantity, unit and direction
Vector Addition In One Dimension
Resultant A vector that can represent two or more other vectors
Graphically Vectors are added by placing the tail of one vector at the head of the other vector.
The resultant is the vector from the tail of the first to the head of the last.
The order of addition does not matter.
Vector Addition in Two Dimensions
Graphically Vectors are added by placing the tail of one vector at the head of the other vector.
The resultant is the vector from the tail of the first to the head of the last.
Addition of Several Vectors
More than two vectors may be added by continuing to place the tail at the head of the last vector.
The resultant is the vector from the tail of the first to the head of the last.
The order of addition does not matter.
Independence of Vector Quantities
Vector Quantities are independent of each other.
6.2 Analytical Method of Vector Addition
Trig Functions
Sin = opposite side/hypotenuse
Cos = adjacent side/hypotenuse
Tan = opposite side/adjacent side
Adding Perpendicular Vectors
The Pythagorean Theorem can be used to calculate the magnitude of the resultant of perpendicular vectors.
R hypotenuse
A Adjacent side
B opposite side
R2 = A2 + B2
Tangent, Sin, or Cos can be used to determine the angle or direction. (Usually the tangent is used.)
Components of Vectors
A vector quantity can be resolved into its perpendicular components.
A = R cos
B = R sin
The choice of axes does not affect the final solution of a problem.
Adding Vectors at Any Angle
Non-perpendicular vectors can be added by resolving each into its vertical and horizontal components and summing them.
1. Resolve the vectors in the vertical and horizontal components.
2. Add the vertical and horizontal components.
3. Use the Pythagorean theorem to find the magnitude of the resultant.
4. Use the Tangent to find the direction of the resultant.
6.3 Applications of Vectors
Equilibrium net sum of all the forces acting on an object is zero.
Equilibrant single additional force that, if applied at the same point as the other forces, will produce equilibrium.
Gravitational Force and Inclined Planes
When solving inclined problems, it is convenient to choose a set of axes perpendicular and parallel to the plane.
