 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that the vector a is equal to the ordered pair minus 2 4 sketch the vectors 2 into vector a minus 1 by 2 into vector a and minus 3 into vector a. We know that if we multiply a vector v by a scalar k then we get a new vector that is vector k v and if the value of k is greater than 0 then direction of vector k v is a long vector v and the magnitude of vector is increased k times and if the value of k is less than 0 then direction of vector k v is opposite to vector v and the magnitude of vector is decreased k times. With this key idea let us proceed to the solution. Here we are given vector a and we have to sketch the vectors that is 2 into vector a minus 1 by 2 into vector a and minus 3 into vector a. So we first draw vector a having components minus 2 4. So taking any point as initial point we move 2 units left 1 2 and 4 units up 1 2 3 4 and reach terminal point of vector in upward direction. So this is vector a. Now we draw 2 into vector a. Now using the key idea here scalar is 2 which is greater than 0 thus direction of 2 into vector a will be along vector a. Now to draw 2 into vector a which is equal to vector a plus vector a thus we will draw vector a twice successively. So first we again draw vector a by moving 2 units left and 4 units up and at this terminal point we again draw vector a by moving 2 units left and 4 units up. So these 2 vectors combined up to make 2 into vector a and we see that its components are minus 4 8. We see that its magnitude is twice of vector a and it has same direction as of vector a. Now we draw minus 1 by 2 into vector a. Here scalar is minus 1 by 2 which is less than 0. So its direction is opposite to vector a and magnitude is half of vector a. So we take any initial point and now we move half of x component of vector a that is 1 unit horizontally and in opposite direction so from left we now move right so we move 1 unit to the right and then we take half of y component of vector a that is 2 units vertically and now in downward direction this is minus 1 by 2 into vector a and its components are 1 minus 2. Now to draw minus 3 into vector a we will draw vector a three times but in opposite direction also minus 3 into vector a can be written as minus of vector a plus minus of vector a plus minus of vector a. So from any initial point we move 2 units right and 4 units down and we reach this point. Now this is minus of vector a from this terminal point we again draw minus of vector a by moving 2 units right and 4 units down and we reach this terminal point and from here again we draw this vector by moving in same direction that is 2 units right and 4 units down we reach this terminal point. Now these three vectors combined to make minus of 3 into vector a and we see its components are 6 minus of 12 thus we have seen how to sketch vectors that is 2 vector a minus 1 by 2 vector a and minus 3 vector a this completes our session hope you enjoyed this session