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Vector Addition

Introduction

      A vector is not just a number. A vector is a number with direction. Even the symbol for a vector is different from that of a number. The symbol for a vector is a capitol letter with a ray above it.

      Two pieces of information are required to describe a vector -- its maginitude (size) and its direction (tilt).

      This information may be stated graphically or algebraically. When stated algebraically vectors may be given in either rectangular form, (x,y), or polar form, r and .

          Rectangular form, (x,y), says:
  • start at the origin,
  • go the required distance in the x direction (the x-component of the vector), then
  • go the required distance in the y direction (the y-component of the vector), then
  • stop, and
  • consider the LENGTH AND DIRECTION OF THE SHORTCUT FROM START TO FINISH as the vector.
          Polar form, r , says:
  • start at the origin,
  • go in the required direction (the direction of the vector),
  • go for the required distance (the magnitude of the vector), then
  • stop, and
  • consider the LENGTH AND DIRECTION OF THE PATH FROM START TO FINISH as the vector.
 

Mental Computation & Resolution of Vectors

      Simultaneously thinking magnitude, direction, x-component, y-component is the key to vector computation. Simultaneously think both polar form and rectangular form.

      The table below provides some simple exercises. Mentally determine the answers. If this is not possible, draw the vector. Press "=" then the other "=" to see the answers.

 
  R (x,y) Answers

1. 5 90° (, )

2. (0,-3)

3. 2 180° (, )

4. 4 (, )

5. (3,0)


      If you wish more practice or to experiment with vectors which do not lie on the axes, see graphing calculator computation, below, and get out your calculator. Graphing calculators have built in functions to change vectors (and complex numbers) from rectangular to polar and polar to rectangular forms.

 

Graphic Vector Addition

      If you understand vector notation, it is often possible to add vectors mentally. Graphic addition permits one to see why mental addition is possible and why algebraic addition through components works so nicely.

      To add vectors graphically, make a head-to-tail trail, then draw a short-cut arrow to connect the start to the finish. The magnitude and direction of the short-cut arrow are the necessary symbols for the answer.

 
To add vectors graphically:
 
1st: Designate an area of the paper to act as the work area for the accurate drawing of the vectors to be added. Remember, lengths should be copied accuretely or scaled accurately. Angles must also be copied accurately but not scaled.
 
2nd: Pick a vector to be first and make an accurate arrow for its length and direction in the work area.
 
3rd: Copy the next vector in the work area PLACING ITS TAIL AT THE HEAD OF THE LAST VECTOR.
 
4th: Repeat the 3rd step for each additional vector to be added.
 
5th: Draw a new vector -- the shortcut vector -- from the tail of the first vector to the head of the last of all the vectors. Be sure to draw an arrowhead at the head of the new vector to indicate the direction of the vector.

      If vectors are not added graphically, they may be added mentally or algebraically. In algebraic computation of the sum of vectors, components are required. Each vector named by maginitude (size) and direction (tilt), polar coordinates, is restated relative to a rectangular coordinate system using, for two-space. x- and y-coordinates, (x,y).


Mental Computation Problems

Add the vectors or state the resultant. Check answers.
1. 5 at 90°, 3 at 90°
2. 4 at 180°, 5 at 90°
3. 4 at 90°, 5 at -90°
4. 7 at 180°, -7 at 180°
5. 2 at 180°, 4 at 0°
6. (2,0), (-4,0)
7. (1,0), (-2,0), (3,0), (4,0)
8. 5 at 40°, 2 at 40°
9. 4 vectors each (7,-2)
10. 1 at 0°, 1 at 90°, 1 at 180°, 1 at 270°
11. 3 vectors each 5 at 90°
12. (0,0), (0,3), (0, -1), (0, 9)
13. (6,-5) + (4,2)
14. (7,3) + (2, -1) - (1, 9)

      For other vector additions, sine and cosine are needed. Take the magnitude and direction of each vector and rewrite each in terms of the horizontal and vertical vectors, the x-component and y-component. Once this is done, all horizontal and vertical vectors are added and the resultant horizontal and vertical vectors are used with the inverse tangent to find the resultant direction of the vector. The horizonal and vertical vectors are then used with the Pythagorean Theorem to find the magnitude of the resultant. CHECK IT OUT IN STEPS BELOW.

      The graphic below dislays work. The table below that will complete computation for the reader.

 

Instructions

The vector addition calculator below uses the same steps as algebraic addition of vectors. THIS CALCULATOR DOES NOT WORK WELL WITH QUADRANTAL DIRECTIONS, vectors which lie on the axes.
Enter the angle in degree measure.
 
1st: Enter the magnitude and direction (in degrees) of each vector or go to the 2nd step.
2nd: Type in the x- and y-component of each vector or
        press "=" to compute the x- and y-component of each vector enter in the 1st step.
Note: Ax is A(cos(theta)), Ay is A(sin(theta)).
3rd: Add the x-components to compute Rx, the x-component of the resultant.
4th: Add the y-components to compute Ry, the y-component of the resultant.
5th: Use (Rx, Ry), the x- and y-components of the resultant to determine the quadrant of the resultant. 5th:
6th: Compute theta, the direction angle, and edit according to the signs of x and y, see step 5.
        Note: tan(theta)= (y/x)
    NOTE: IF THE X-COMPONENT IS ZERO,
  • DO NOT COMPLETE THE COMPUTATION BY COMPUTER.
  • y/x is y/0, division by zero is undefined.
  • If (0,+), the angle is 90° and if (0,-) the angle is 270°.
7th: Compute R, the magnitude of the resultant. Note: R = (Rx2 + Ry2)
8th: Round as required in the situation.
 
 
 
Vector addition calculator -- Scroll up for instructions.
Vector Name   Magnitude, V   Direction, theta   x is V(cos(theta)) y is V(sin(theta))
A 1st: 1st:
B 1st: 1st:
C 1st: 1st:
D 1st: 1st:
Resultant
8th: Round as required in the situation.

Mental Computation Problems

Answer the question
1. How do you subtract vector B from vector A? Check answer.
2. Scalar multiplication may be used to complete problems 9 and 11.
Explain a computation short-cut used to complete these problems. Check this link to see if you used scalar multiplication to complete the repeated additon.

Hand-held Graphing Calculator Computation of Resultant

      In math, vectors are written as (x,y) or r .

      On the TI-85, -86 or -89, they are written as [x,y] or [r ].

      The symbols [ and ] are above the symobls ( and ), and the symbol is above the , symbol. Use the shift key to print them.

      Two TI calculator functions facilitate vector computation at the elementary level. They are "into rectangular," Rec, and "into polar," Pol, and may be stored in the user's CUSTOM menu.

      Type [2,0] + [5 180] and press [ENTER].

      If the result is [-3,0] and you wish polar notation, press Pol. 180° and press [ENTER].

      Type [2,0] + [5 180° and press [ENTER].


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        Scalar multiplication is used to multiply a vector by a nonvector number.
        If a is a scalar and (x,y) is a vector, then a(x,y) is (ax,ay).
ex. 1. 5 at 90, 5 at 90 is 2(0,5) or (0,10) or 10 at 90°.
ex. 2. 4 vectors each (7,-2) is 4(7,-2) is (28,-8) or 29.12 at 105.945°.
ex. 3. 3 vectors each 5 at 90° is 3(0,5) or (0,15) or 15 at 90°.


































        Add the opposite.
(4,6) - (2,5) = (4,6) + (-2,-5) = (2,1).