Find the magnitude and direction of resultant of two vectors analytically. Discuss the special cases for angle between the vectors: (i) 0° (ii) 180° (iii) 90°.

Let two vectors space straight P with rightwards arrow on top space and space straight Q with rightwards arrow on top be represented by two adjacent sides OA and OB of parallelogram OACB.
    From C draw CD perpendicular to OA produced.
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#6 {main}</pre>       [corresponding angle]
From increment space CAD comma
              AD space equals space AC space cos space straight theta space equals space straight Q space cosθ
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#6 {main}</pre>

Let two vectors  be represented by two adjacent sides OA and OB of

Special cases:

Let two vectors  be represented by two adjacent sides OA and OB of

Let two vectors  be represented by two adjacent sides OA and OB of

                 

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Two vectors straight A with rightwards arrow on top space and space straight B with rightwards arrow on top are represented by the two sides of a parallelogram drawn from point O. Show that the diagonal not passing through O is given by plus-or-minus left parenthesis straight A with rightwards arrow on top minus straight B with rightwards arrow on top right parenthesis.


Let two vectors straight A with rightwards arrow on top space and space straight B with rightwards arrow on top be represented by two adjacent sides of parallelogram OP and OQ drawn from O as shown in the figure.
Complete the parallelogram OPSQ.

Let two vectors  be represented by two adjacent sides of parallelog
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#6 {main}</pre> not passing through O represent the difference of two vectors.

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State and prove associative law for vector addition.


Statement: The associative law states that the vector addition is same, in whatever grouping they are added.

Statement: The associative law states that the vector addition is sam

Statement: The associative law states that the vector addition is sam

               

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State the properties of vector addition.

The properties of vector addition are: 

(i) A vector can be added only to a vector. 

(ii) Closure property of addition: The sum of two vectors is also a vector. Hence, vectors are closed under addition.

(iii) Vector addition is commutative.
 
i.e. 

(iv)Vector addition is associative. 
 
i.e.
 

(v) Vector addition is distributive.

i.e. 
 

(vi) Magnitude of the resultant of two vectors is less or equal to the sum of the magnitude of two vectors and greater or equal to the magnitude of the difference of the magnitude of two vectors. 

i.e. 

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#6 {main}</pre>?


The magnitude of the resultant of two vectors space straight P with rightwards arrow on top space and space straight Q with rightwards arrow on top acting at angle straight theta is given by
                      space space straight R equals square root of straight P squared plus straight Q squared plus 2 PQcosθ end root
Here,       P = A + B,        Q = A - B

The magnitude of the resultant of two vectors  acting at angle  i
                                  
                    

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