 Hello and welcome to the session. In this session we will discuss how to represent scalar multiplication graphically by scaling vectors and possibly reversing their direction. We will also perform scalar multiplication component wise and will compute magnitude and direction of scalar multiple. First of all we shall discuss scalar multiplication multiplying a vector with a number is called scalar multiplication the number is called scalar because it does not have a direction. Now the product of a scalar c and a vector that is a vector is a vector with the same direction as vector a and its magnitude will be c into magnitude of vector a where c is greater than 0 and if c is less than 0 then we take its positive value for magnitude. We can denote magnitude or length of vector a by magnitude of vector a that is by taking absolute value of vector a or norm of vector a or simply a. For example if we have any vector a and we multiply this vector with a scalar 2 then we will get a new vector say vector b which is equal to twice of vector a and its magnitude that is magnitude of vector b will be equal to 2 into magnitude of vector a. Now if the scalar c is negative that is suppose we multiply vector a by scalar minus of 2 then we get a new vector that is vector b which is equal to minus 2 into vector a that is minus 2 vector a it means the resulting vector will be in opposite direction to the given vector a because here scalar is equal to minus 2 which is less than 0 also its magnitude will be absolute value of minus 2 into magnitude of vector a which implies that magnitude of vector b will be equal to 2 into magnitude of vector a because absolute value of minus 2 is equal to 2. Now let us see it with the help of a figure. Now if we have any vector a that is this is vector a now here we see that twice of vector a is equal to vector a plus vector a that is at terminal point of vector a we have again drawn vector a in the same direction and equal magnitude now see this resulting vector that is vector b which is equal to twice of vector a and its magnitude is twice the length of vector a and if we take opposite direction then the resulting vector is minus 2 vector a and its magnitude is also twice the length of vector a. Let us examine the magnitude and direction of minus of 1 by 3 into vector a see here we have multiplied vector a with scalar minus of 1 by 3 so here c is equal to minus of 1 by 3 which is less than 0 thus the direction of this vector will be opposite to vector a and its magnitude will be 1 by 3 into magnitude of vector a which means its length will be one third of the length of the given vector a. Now we shall discuss scalar multiplication component wise now suppose we have vector a which is given by the ordered pair 1 3 and we have to find 2 into vector a we know that twice of vector a is given by vector a plus vector a so this is equal to the ordered pair 1 3 plus the ordered pair 1 3 which is equal to the ordered pair 1 plus 1 3 plus 3 which is equal to the ordered pair 2 6 so 2 into vector a is given by the ordered pair 2 6 in other words we say that we multiply each component of vector a by scalar 2 that is 2 into vector a is equal to 2 into the ordered pair 1 3 which is equal to the ordered pair 2 into 1 2 into 3 that is equal to the ordered pair 2 6 thus in general if we have vector with components x y then scalar multiplication c into ordered pair x y will be equal to the ordered pair c x c y where c is greater than 0 or less than 0 now we are going to discuss magnitude of component wise scalar multiplication suppose we have vector a which is equal to the ordered pair 6 8 and if we want to find magnitude of minus 3 into vector a which is denoted by the absolute value of minus 3 into vector a or norm of minus 3 into vector a first of all let us find magnitude of vector a which is given by square root of 6 square plus 8 square and this is equal to square root of 36 plus 64 that is equal to square root of 100 which is equal to 10 so magnitude of vector a is equal to 10 now let us find minus 3 into vector a which is given by minus 3 into ordered pair 6 8 that is equal to the ordered pair minus 3 into 6 minus 3 into 8 which is equal to the ordered pair minus 18 minus 24 so minus of 3 into vector a is given by the ordered pair minus 18 minus 24 now we find its magnitude so magnitude of minus 3 into vector a is equal to square root of minus 18 whole square plus minus 24 whole square and this is equal to square root of minus 18 whole square that is 324 plus minus 24 whole square that is 576 and this is equal to square root of 900 which is equal to 30 so magnitude of minus 3 into vector a is equal to 30 which is equal to 3 into 10 that is equal to 3 into magnitude of vector a thus we have magnitude of minus 3 into vector a is equal to 3 into magnitude of vector a or we can write it as absolute value of minus 3 into magnitude of vector a therefore we have magnitude of c into vector a is equal to absolute value of c into magnitude of vector a therefore we say that length of vector a is increased c times if c is greater than 1 and length is decreased c times if 0 is less than c is less than 1 now we will discuss scalar multiplication graphically now let us take vector a which is given by the ordered pair 3 1 now let us draw it on the coordinate plane with any initial point we move 3 units right and 1 unit up and we reach this terminal point with arrow showing upward direction this is vector a now we want to find 2 into vector a since 2 into vector a is equal to vector a plus vector a so again at its terminal point draw vector a again in same direction that is moving 3 units right and 1 unit up so we get vector 2 a in same direction from this initial point we have moved 1 2 3 4 5 6 6 units to right and 2 units up so this vector is given by the ordered pair 6 2 which is twice the magnitude of vector a and in the same direction as scalar c is greater than 0 here c is equal to 2 and it is greater than 0 thus in this session we have learnt how to represent scalar multiplication graphically by scaling vectors and possibly reversing their direction and we have also learnt how to perform scalar multiplication component wise and how to compute magnitude and direction of scalar multiple this completes our session hope you enjoyed this session