Vectors
-
Finding the Resultant for 2 Vectors (A&B) (Triangle
Method)
1) Set up a scale (ex. 1 N = 2 cm.)
2) Convert the magnitudes of your vectors into lengths
(ex. 7 N => 7x2= 14cm.)
3) Select a starting point.
4) Draw the first vector from the starting point.
5) Starting at the arrow tip of the first vector, draw
the second vector.
6) Draw a line to connect the original starting point
to the arrow tip of the second vector.
7) Measure the length of that line (the resultant) and
convert from length to magnitude.
8) Measure the angle of the resultant from a definite
point of reference.
NOTE: If one (or both) of the vectors is not along
the horizontal or vertical axis, you will need to resolve that vector into
its vertical and horizontal components. Then solve for the resulting
right triangle.
Mathematical Relationships in Triangles
1) For Right Triangles ONLY:
=> c2 = a2 + b2
=> sin B = b/c
so ... b = c * sin B
=> cos B = a/c
so ... a = c * cos B
2) For ANY Triangle:
=> c2 = a2 + b2 - 2 * a * b * cos C
Note: if angle C is > 90o, then the cos C is equal to the negative
of the complement of angle C
[for C > 90o: cos C = - cos (180 - C)]
=> a =
b = c
sin A sin B
sin C
Resolving Vectors into Components
-
The easiest and most common method is to resolve a vector
into its vertical and horizontal components. This is done by:
1) Draw the vector at its scale length and in the proper
direction.
2) Draw two vectors to make a right triangle, with the
original vector as the hypotenuse.
3) Using one of the acute angles of the triangle and
the relationships given above, solve for sides a and b, the horizontal
and vertical sides.
|
