Vector operations



Vector operations

Addition of vectors: 

If two scalars are added resulting scalar will be unique which will be equal to sum of the given two scalars. For examples
if two scalars 2 and 8 are added their sum will be always  equal to 2+8 =10.
But, the addition of vectors is complicated. If we add two vectors of magnitudes 2 and 8 the resultant vector’s magnitude will be 6 or 10 or any value  between 6 and 10 depending on the directions of the vectors we are adding.

  1. If the two given vectors are acting in same direction then the magnitude of the resultant vector will be 10 units,
  2. If the two given vectors are acting in opposite directions then the magnitude of resultant vector will be 6 units,
  3. If the two given  vectors are acting in different directions then the magnitude of the resultant vector will be between 6 and 10.

How to add two given  vectors(Geometrical representation): 

Suppose overline{AB} and overline{CD} are two given vectors.
i)If the two vectors are acting in the same direction,take the first vector  Suppose overline{AB}, to the terminal point of overline{AB} connect the initial point of overline{CD}.
We get overline{AB} + overline{CD} = overline{AD}.
The magnitude of resultant vector will be equal to the sum of magnitudes of overline{AB} and overline{CD}
i.e When two vectors are acting in same direction thesum of the magnitudes of the vectors = Magnitude of resultant vector”
AB+CD = AD

ii)If the two vectors are acting in different directions:
In that case the procedure of addition will be same but the direction and magnitude of resultant vector will be different.Suppose overline{AB} and overline{CD} are two given vectors acting in different directions as shown in the below fig(a).
To add these two vectors connect the initial point C of IInd vectors overline{CD} to the terminal point B of First  vector overline{AB}.Now join the initial point A of first vector overline{AB} with the terminal point D of the second vector overline{CD}.The vector overline{AD} taken in reverse order overline{AD} (closing side taken in reverse order) represents the resultant vector both in magnitude and direction.
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i.e. When two vectors are acting in different directions the “Sum of the magnitudes of the vectors > Magnitude of resultant vector”.

Laws of vector addition:

i) Vector addition is commutative: bar{a} + bar{b} = bar{b}bar{a}
Proof:
Suppose overline{OA} = bar{a} and overline{AB} = bar{b} are two given vectors.Now let us add these two vectors. To add
them let us connect the initial position of overline{AB} to the terminal point of overline{OA} in anti clock wise direction,now closing side OB taken in the reverse order represents the resultant vector bar{r} .
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Now, draw a vectors parallel to bar{a} and bar{b} and complete the parallelogram OABC.From the Fig(ii) overline{CB} = overline{OA} = bar{a} and overline{AB} = overline{OC} = bar{b}.
From triangle OAB overline{OA} + overline{AB} = overline{OB}
i.e bar{a} + bar{b} = bar{r} – – – – – – – – – – – – (1)
From triangle OCB overline{OC} + overline{CB} = overline{OB}
i.e bar{b} + bar{a} = bar{r} – – – – – – – – – – – – (2)
from eq(1) and (2) we get bar{a} + bar{b} = bar{b} + bar{a}
Hence vector addition is commutative.

ii) Vector addition is associative:
Let overline{OA} = bar{a},overline{AB} = bar{b} and overline{BC} = bar{c} be three different vectors.
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To add the given three vectors bar{a} , bar{b} and bar{c} we have to connect the initial point of overline{AB} to the terminal point of overline{OA} and the initial point of overline{BC} to terminal point of overline{AB}.The closing side taken in reverse order represents the resultant vector.
Hence resultant overline{OC} = bar{r}.

Proof:
To add three vectors we have to first add two vectors and to sum vector we will add the third vector.
Form  fig(ii) from the triangle OAB overline{OA} + overline{AB} =overline{OB} = (bar{a} + bar{b}) – – – (1)
from triangle OBC overline{OB} + overline{BC} = overline{OC} = bar{r} – – – – – – – – – – – – (2)
substitute the value overline{BC} and overline{OB} = (bar{a} + bar{b}) from eq(1) to eq(2)
we get (bar{a} + bar{b}) + bar{c} = overline{OC} = bar{r} – – – – –  -(A)
From triangle ABC overline{AB} + overline{BC} = overline{AC}
i.e (bar{b} + bar{c}) = overline{AC} – – – – – – – – – (3)
From triangle OAC overline{OA} + overline{AC} = overline{OC} = bar{r} – – – – – – – – (4)
solving eq(3) and eq(4) we get bar{a} + (bar{b} + bar{c}) = overline{OC} = bar{r} – – – – – – – (B)
Comparing eq(A) & eq(B) we get (bar{a} + bar{b}) + bar{c} = bar{a} + (bar{b} + bar{c})
Hence,vector addition is associative.

Subtraction of Vectors: 
Subtraction of vectors is also a form of addition.Addition of two vectors acting in opposite direction is called subtraction of vectors.
Suppose as in fig(i) overline{AB} = bar{a} and overline{CD} = bar{b} are two vectors, to subtract bar{b} from bar{a} we have to add the negative vector of bar{b} to bar{a}, i.e bar{a}bar{b} = bar{a} + ( –bar{b}).
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In the fig (ii) we have drawn overline{BE} = (-bar{b}) negative vector of overline{CD} = bar{b} , now connect the initial point of
latex overline{BE}$ to the terminal point of overline{AB}. From the resultant vector of addition of these two vectors is overline{AE}.
Therefore, overline{AB} + overline{BE} = overline{AE}
i.e bar{a} + (- bar{b}) = bar{a}bar{b} = overline{AE}.
Note:-*Subtraction of vectors is not commutative i.e. bar{a}bar{b}not =bar{b}bar{a}
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