Vectors - Summary

  1. Vector vs. Scalar
    1. Scalars have magnitude only (mass, time, distance, speed, etc.)
    2. VECTORS have a magnitude and a direction (displacement, velocity, acceleration, force, etc.)
    1. A vector quantity is represented by a scaled arrow-tipped line segment called a RAY.

  2. VECTOR COMPOSITION (Vector Addition)
    1. Vectors are added head to tail (ie. the next vector begins where the previous vector ends)
    2. Vectors may be added in any order (vector addition is commutative)
    3. The vector sum or RESULTANT, is drawn from the tail of the first vector drawn to the head of the last vector drawn. Its magnitude is determined by measuring its length and applying it to the scale. Its direction is found using a protractor and measuring its angle from the origin.
    4. When added vectors produce a resultant; however, each vector still produces its full effect on the object in question. (Vectors act independently of each other.) The resultant merely indicates the combined effect of all the vectors acting on an object.

  3. VECTOR RESOLUTION (Vector Division)
    1. Any single vector can be broken down into 2 perpendicular COMPONENT vectors.
    2. x-component (horizontal, east-west) and y-component (vertical, north-south)
    3. To resolve a vector into its perpendicular components form a right triangle with the original vector being the hypotenuse of the triangle. The two sides of the triangle are the perpendicular components. Measure the length of the sides to determine the magnitude of the components and place the arrows in the appropriate direction.
    4. It is possible to adjust the magnitudes of the components by adjusting the angle of the original vector. The smaller the angle between the original vector and a component the greater the magnitude of that component.

  4. Solving Vector Problems Analytically (Mathematically)
    1. Vector problems are solved using trigonometry for right triangles.
    2. To resolve a vector into 2 components use the following guidelines:
      3.       
      1. must always be stated relative to the positive x-axis.
      2. the "x" component is found using the cosine function in the form x = V cosq (where "V" stands for the original vector)
      3. the "y" component is found using the sine function in the form y = V sinq (where "V" stands for the original vector)
    3. To find the resultant of 2 vectors acting at right angles to each other use the following guidelines:
      4.       
      1. The magnitude of the resultant can be solved using the Pythagorean theorem c2 = a2 + b2 in the form R2 = x2 + y2 (where "R" is the resultant, "x" is the vector on the x-axis, and "y" is the vector on the y-axis).
      2. The direction of the resultant can always be found using the tangent function in the form Tan q = y/x. The compass directions are determined by placing the y-axis direction relative to the x-axis direction.