 Hello and welcome to the session. In this session we will discuss algebraic vectors, how to add vectors component-wise and by parallelogram rule and we will understand that magnitude of some of two vectors is typically not the sum of magnitudes. First of all let us discuss algebraic vectors. Now vectors can be represented as definitely by using object pairs of real handlers. For example, object pair 2,3 will represent vector in standard position. Now here you can see in this type of object pair we are using angled brackets to represent a vector. Now from this object pair we understand that this is the vector whose initial point is at origin and terminal point is at the point 2,3. Now the terminal plane we have represented this vector like this. Here you can see its initial point is at the origin and terminal point is given by the object pair 2,3. Now from this representation we can see that horizontal distance is 2 units and vertical distance is 3 units. So we can say that this vector is the resultant of a horizontal vector with a magnitude of 2 units and a vertical vector with a magnitude of 3 units. Now since the vectors with same magnitude and direction are equal therefore so many vectors can be represented by the same object pair 2,3. The initial point can be any point in the plane. Now here you can see this is also a vector which is represented by the object pair 2,3 and this is again a vector that is represented by the object pair 2,3. Because these all have same horizontal magnitude that is 2 units, same vertical magnitude that is 3 units from their initial points. Now let us discuss addition of vectors. Suppose we have vector AB which is equal to object pair 4,3 and vector BC is equal to object pair 2,5. Now we have to find components of vector AC. First of all let us draw these vectors on the coordinate plane. Now x component of vector AB is 4 and y component is 3. So for drawing vector AB first of all we take any initial point A on this coordinate plane and let this be point A. Now from this initial point A we move 1,2,3 and 4 units to the right and 3 units upwards and we reach at this point and this is the terminal point B. So we have drawn vector AB. Now let us draw vector BC whose x component is 2 and y component is 5. Now the terminal point of vector AB will be the initial point of vector BC. So this point B is the initial point of vector BC and from this point we move 2 units to the right and 5 units upwards and we reach at this point and this is the point C that is the terminal point C of vector BC. So we have drawn vector BC. Now we have to find components of vector AC. Now we know that vector AC has initial point A and terminal point C and as we have moved in up the direction so we will show it by this arrow. Now here these arrows show the directions of these vectors. Now here you can see that joining the points A,B and C we get a triangle that is triangle ABC. Now to find components of AC here we see that from its initial point that is the point A we have moved 4 plus 2 that is 1 2 3 4 5 and 6 units to the right and 3 plus 5 that is 1 2 3 4 5 6 7 and 8 units upwards and reached its terminal point C. So x component of vector AC is given by 4 plus 2 and its y component is given by 3 plus 5. So this is equal to order pair 6 8. Now we see that vector AB plus vector BC is equal to order pair 4 3 plus order pair 2 5 which is equal to order pair 4 plus 2 3 plus 5 that is equal to order pair 6 8 which is equal to vector AC thus vector AB plus vector BC is equal to vector AC and this is also called triangle method of vector addition. Now let us discuss parallelogram rule of vector addition. Now see we have drawn this parallelogram and we have vector OA is equal to order pair 4 1 as here you can see initial point of vector OA is O and its terminal point is A. So from this initial point we have moved 1 2 3 and 4 units to the right and 1 unit up and we have reached at its terminal point A. So vector OA is equal to order pair 4 1 similarly vector OB is equal to order pair 4 2 4 and we have to find resultant vector OC. Now initial point of vector OC is O and its terminal point is C. Now starting from the initial point that is to region with coordinates 00 we have moved 1 2 3 4 5 and 6 units to the right and 1 2 3 4 and 5 units up to reach its terminal point C. So vector OC is equal to order pair 6 5 thus we see that vector OA plus vector OB is equal to order pair 4 1 plus order pair 2 4 that is equal to order pair 4 plus 2 1 plus 4 which is equal to order pair 6 5 that is equal to vector OC and this is called parallelogram rule of vector addition. Now this rule is derived from triangle rule only. Now here you can see that in this parallelogram OA C is a triangle and vector AC is same as vector OB because here also from the initial point A we have moved 2 units to the right and 4 units up to reach the terminal point C. So using triangle rule of addition we have vector OA plus vector AC is equal to vector OC. Now if vector A is given by order pair x 5 then its magnitude is given by magnitude of A is equal to square root of x square plus y square and magnitude is always positive. Now vector AB is given by order pair 4 3 so its magnitude will be equal to square root of 4 square plus 3 square that is equal to square root of 16 plus 9 which is equal to square root of 25 that is equal to 5. Similarly we can find magnitude of vector BC that will be equal to square root of 2 square that is 4 plus 5 square that is 25 so this is equal to square root of 29. Now let us find magnitude of vector AB plus vector BC which is equal to magnitude of vector AC. Now vector AC is given by order pair 6 8 so this is equal to square root of 6 square plus 8 square that is equal to square root of 36 plus 64 which is equal to square root of 100 that is 10. Now here you can see 5 plus square root of 29 is not equal to 10 this means magnitude of vector AB plus magnitude of vector BC is not equal to magnitude of vector AC. Thus magnitude of sum of two vectors is typically not the sum of the magnitudes. Now let us see how to know direction of the resultant vector when we add two vectors. In triangle rule suppose we have to add two vectors A and B. Here we have drawn the two vectors in the given direction. The terminal point of vector A will be the initial point of vector B and in same direction then join the initial point of vector A and terminal point of vector B. This is the resultant vector C which is equal to vector A plus vector B and its initial point coincides with initial point of vector A and its terminal point coincides with terminal point of vector B and in parallelogram rule the initial point of the resultant vector and the other two vectors coincide and we draw an arrow from the common initial point to the opposing corner of the parallelogram and this will give us the direction. So in this session we have discussed algebraic vectors and triangle and parallelogram rule of vector addition and this completes our session. Hope you all have enjoyed the session.