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Vectors

Vector
has magnitude and direction
Resultant vector
"sum" of several vectors; the effect of the resultant vector is always greater than the effect of its individual components.
Concurrent vectors
act on the same point at the same time
Equilibrant vector
a vector that produces equilibrium; it is equal in magnitude and opposite in direction to the resultant vector

Graphical method of vector addition:

  1. Vectors are represented graphically by using arrows. The length of the arrow represents the vector’s magnitude; the direction the arrow points represents the direction of the vector. Vectors are drawn graphically using a scale.
  2. Vectors are added graphically by placing them "tips to tails." The tail of the second vector is touches the tip of the first vector, etc.
  3. The resultant vector is drawn graphically by placing the tail of the resultant at the tail of the first vector and the tip of the resultant at the tip of the last vector. It is draw from where you started to where you ended.
  4. The resultant vector’s magnitude can be determined graphically by measuring its length and converting its length using the scale chosen. Its direction is the direction that it points.

There are two parts that describe the resultant vector graphically—its magnitude and its compass direction.

Inertial Frame of Reference

A frame of reference in which Newton's law of inertia is valid. In an inertial frame of reference, acceleration is zero. An object moves at a constant velocity or is at rest. Remember, that an object at rest relative to one frame of reference, is not necessarily at rest relative to another. For example, if a ball is thrown up in a bus and lands at your feet. The bus is either stopped or is moving at constant speed.

Relative velocities

The velocity of an object is relative to the observer who is making the measurement. For example, a hitchhiker may see two cars pass simultaneously at 35 m/s. But, relative to the cars, the speed of each is zero.

Properties of vectors

  1. A = B
    if the magnitude and direction of each are the same
  2. C = A + B
  3. A - B = A + (-B)
  4. A * B = AB cos q
    This is read as the "dot product" of A and B. In other words, the product of two vectors is a scalar quantity.

    Mathematical method of vector addition:

    1. One dimension:
      • Vectors that act in a line linearly are assigned positive and negative signs to indicate their direction. Positive signs are assigned to vectors acting right, up, east, or north. Negative signs are assigned to vectors acting left, down, west, or south.
      • Using their signs, vectors are added algebraically to determine the magnitude and sign (direction) of the resultant.
    2. Two dimensions:
      • The resultant vector is found mathematically. We will use the component method of vector addition. A resultant vector can be considered to be the vector sum of its resultant x-component and its resultant y-component, separated by 90°. This can also be done using graphing calculators.
      • The magnitude of the resultant vector can be determined using the Pythagorean theorem (c2 = a2 + b2) or using the graphing calculator. On the TI-83, the keystrokes are: 2nd angle, R>Pr(. Enter the magnitude of the x-component and then the magnitude of the y-component, separated by a comma. End with a parenthesis. Enter, displaying the magnitude of the resultant vector.
      • The direction of the resultant vector can be expressed as an angle between 0° and 90° and a compass direction. Sketch the x- and the y-components. Draw the resultant vector following the graphical method. The angle is the angle between the tail of the first vector and the tail of the resultant vector. The angle is found mathematically using the tangent function (tangent of the angle is equal to the opposite side over the adjacent side). The compass direction is described as "direction of the resultant vector" of "direction of the first vector." For example, if the x-component points west and the y-component points south, the resultant would be south of west. In this solution, there are three parts to your answer: magnitude, angle, and compass direction.
      • The direction of the resultant vector can be determined using the graphing calculator to determine the angle. Again, sketch the x-component and the y-components, drawing the resultant vector according to the graphical method. On the TI-83, the keystrokes are: 2nd angle, R>Pq. Enter the magnitude of the x-component and then the magnitude of the y-component, separated by a comma. You must use a positive or negative sign to indicate the quadrant. (For example: The x-component is 4 N, east. It would be entered as +4. The y-component is 6 N, south. It would be entered as -6). End with a parenthesis. Enter, displaying the angular direction of the resultant vector.
      • If you do not want to express the angular direction as a number between 0° and 90° with an appropriate compass direction (written as resultant of first), you may express the resultant as magnitude and an angle between 0° and 360°, expressing the angle in the form appropriate for the quadrant in which it occurs. In this method, all angles are relative to the positive x-axis.

    20 N, 40°, north of east is equivalent to 20 N, 40° graphing calculator answer is 20 N, 40°
    20 N, 40°, north of west is equivalent to 20 N, 140° graphing calculator answer is 20 N, 140°
    20 N, 40°, south of west is equivalent to 20 N, 220° graphing calculator answer is 20 N, -140°
    20 N, 40°, south of east is equivalent to 20 N, 320° graphing calculator answer is 20 N, -40°
    20 N, 50°, east of north is equivalent to 20 N, 50° graphing calculator answer is 20 N, 50°
    20 N, 50°, west of north is equivalent to 20 N, 130° graphing calculator answer is 20 N, 130°
    20 N, 50°, west of south is equivalent to 20 N, 230° graphing calculator answer is 20 N, -130°
    20 N, 50°, east of south is equivalent to 20 N, 310° graphing calculator answer is 20 N, -50°

    Resolution of vectors
    a method of finding the horizontal and vertical components of a resultant vector that "add" to yield a resultant having that magnitude and direction
    1. Sketch the resultant vector. Draw its x-component and its y-component. (remember to draw the components tips to tails!)
    2. Use the appropriate trig functions to determine the magnitudes of the components. Determine the compass directions of the components from the sketch.

    Sin q = (opposite side/hypotenuse)

    Cos q = (adjacent side/hypotenuse)

    Interactive Vector Addition Vector Addition

    Advanced Calculations

    To "add" vectors, resolve each into their x and y components. Assign signs to the components corresponding to their quadrant location. Add all vectors in the x direction (this is your resultant x vector). Add all vectors in the y direction (this is your resultant y vector). The resultant vector is the square root of the sum of the squares of the resultant x and the resultant y vectors. The angle is the cotangent of the ratio of the resultant y and the resultant x vectors.

    Static equilibrium

    Equilibrium occurs when there is no acceleraion (v=0 or v=constant) and there is no unbalanced force (the sum of the forces is zero). An object can be in equilibrium in one direction, but not necessarily in equilibrium in another direction.

    Unit vector

    A unit vector has only one purpose - to specify a direction (i, j, k in the x, y, z plane. i, j, k have a magnitude of unity, have no dimensions, and carry no units. The usefulness of unit vectors is that other vectors can be expressed in terms of them.

    Vectors Sample Problems

    Vectors Homework