 Hello and welcome to the session. In this session we will discuss about the vectors, a quantity that has magnitude as well as direction is called a vector. This is a directed line segment which is denoted as vector a b or simply we can say vector a and this point a from where the vector starts is called the initial point and the point b where the vector ends is called the terminal point and the distance between the initial and terminal points of a vector is called the magnitude of the vector which is denoted as this or this or we can simply write a. Next we should discuss about the position vector. Consider a point p in space with coordinates x, y, z with respect to the origin o, 0, 0, 0. Then this vector o, p with o as the initial point and p as the terminal point is called the position vector of the point p with respect to the origin o that is the position vector of point p is given as vector o p equal to vector r equal to x i cap plus y j cap plus z k cap and the magnitude of the vector o p is given by square root x square plus y square plus z square. Next we have direction cosines, alpha is the angle made by the vector r with the positive direction of x axis, beta is the angle made by the vector r with the positive direction of y axis and in the same way gamma is the angle made by the vector r with the positive direction of z axis. So these angles, alpha, beta, gamma are the direction angles and the cosine values of these angles that is cos alpha cos beta cos gamma which are also denoted by l m n r d direction cosines of vector r. We have cos alpha is equal to x upon r where r is the magnitude of vector r. So from here we get x is equal to l r since l is cos alpha. Then we have cos beta is equal to y upon r. This gives us y is equal to m r and cos gamma is equal to z upon r. This gives us z is equal to n r. So the coordinates of point p are given by l r m r n r. The numbers l r m r n r proportional to the direction cosines are called the direction ratios of vector r and they are denoted by a b c. So these are the direction ratios of vector r. So from here we get a very important relation. l is equal to a upon r, m is equal to b upon r and n is equal to c upon r. This is a very important relation. Next is the types of vectors. A zero vector is one in which the initial and terminal points coincide. This is also called null vector. It is denoted by this. A zero vector cannot be assigned a definite direction as it has zero magnitude. A vector like vector a a or a vector b b represent the zero vector. The next is the unit vector. Unit vector is one whose magnitude is unity. A unit vector in the direction of a given vector a is denoted by a tap. Then we have co-initial vectors. Two or more vectors having same initial point are called co-initial vectors. Then collinear vectors, two or more vectors are said to be collinear if they are parallel to the same line irrespective of the magnitudes and directions. The next is equal vectors. Two vectors a and b are said to be equal if they have the same magnitude and direction regardless of the position of the initial points and they are written as vector a equal to vector b. Next is the negative of a vector. A vector whose magnitude is same as that of a given vector be it vector a b but the direction is opposite to that of the given vector. So if we are given a vector a b then negative of vector a b is b a. Next we discuss about the addition of vectors. A vector a b simply means the displacement from a point a to the point b. If you consider this triangle a b c now let displacement from point a to point c given by vector a c would be equal to the displacement from point a to point b that is vector a b plus the displacement from point b to point c that is vector b c. So when we add two vectors we see that the initial point of one coincides with the terminal point of the other. This is the triangle law of vector addition. Now when we take the sum of the three sides of this triangle in order that is we have vector a b plus vector b c plus vector c a their sum is equal to zero vector. If the two vectors a and b are represented by two adjacent sides of a parallelogram o a c b in magnitude and direction then their sum that is a plus b is represented in magnitude and direction by the diagonal of the parallelogram through their common point. That is the side o a of the parallelogram is the vector a side o b of the parallelogram is the vector b and the diagonal o c of the parallelogram o a c b is given by the sum of the two sides that is vector a plus vector b. This is called the parallelogram law of vector addition. Next we have the properties of vector addition. The first property says for any two vectors a and b vector a plus vector b is equal to vector b plus vector a this is called the commutative property of vector addition. Then next we have for any three vectors a b c vector a plus vector b plus vector c is equal to vector a plus vector b plus vector c and this is called the associative property of vector addition. Next we discuss about the components of a vector the position vector of the point p with reference to the origin o is given by vector o p equal to vector r equal to x i cap plus y j cap plus z k cap this form of any vector is called the component form of the vector and here x y z are called the scalar components of r and x i cap y j cap and z k cap are called the vector components of vector r and whenever we have a vector a lambda be any scalar then the product of the vector a by the scalar lambda is given by lambda vector a this is also a vector collinear to vector a now the magnitude of the vector lambda a is equal to modulus lambda into magnitude of vector a now addition of vectors and multiplication of a vector by a scalar together give the following distributive laws according to which we have if a and b be any two vectors k and m be any scalars then k multiplied by vector a plus m multiplied by vector a is equal to k plus m multiplied by vector a whereas we know that k and m are scalars the next property says k multiplied by m into vector a is equal to k m multiplied by vector a the next one is k multiplied by vector a plus vector b is equal to k multiplied by vector a plus k multiplied by vector b and also one very important point is that the unit vector of vector a is given by one upon magnitude of vector a into vector a suppose we have two vectors vector a equal to i cap plus 3 j cap and vector b equal to 2 i cap plus 4 j cap then vector a plus vector b is equal to i cap plus 3 j cap plus 2 i cap plus 4 j cap equal to 3 i cap plus 7 j cap let the sum of these two vectors be given by vector c now if we need to find the unit vector in the direction of the sum of these two vectors that is we will find the unit vector in the direction of the vector c so for this what we do we find the magnitude of vector c which is given by square root of 3 square plus 7 square that is equal to square root of 58 so the required unit vector c cap is given by one upon magnitude of the vector c into the vector c that is equal to 1 upon square root 58 into 3 i cap plus 7 j cap that is c cap is equal to 3 upon square root 58 i cap plus 7 upon square root 58 j cap next we discuss about the vector joining two points if p1 with coordinates x1, y1, z1 and p2 with coordinates x2, y2, z2 are any two points then the vector joining the points p1 and p2 is the vector p1 p2 which is given by vector op2 minus vector op1 from the triangle law of vector addition and this is equal to x2 minus x1 i cap plus y2 minus y1 j cap plus z2 minus z1 k cap this is the vector p1 p2 now the magnitude of vector p1 p2 is given by square root of x2 minus x1 the whole square plus y2 minus y1 the whole square plus z2 minus z1 the whole square suppose we have two points p with coordinates 2 3 0 and q with coordinates minus 1 minus 2 minus 4 then the vector joining the points p and q directed from p to q is given by pq equal to minus 1 minus 2 i cap plus minus 2 minus 3 j cap plus minus 4 minus 0 k cap this is equal to minus 3 i cap minus 5 j cap minus 4 k cap this is the vector pq next we discuss the section formula we have two points p and q position vector of the point p is vector a and position vector of point q is vector b then the position vector of point r which divides the line segment joining the points p and q internally in the ratio m is to n is given by n into vector a plus m into vector b upon m plus n and now the position vector of the point r dividing the line segment joining the points p and q externally in the ratio m is to n is given by m into vector b minus n into vector a upon m minus n we are given the position vector of point p as i cap plus 2 j cap minus k cap and position vector of the point q as minus i cap plus j cap plus k cap and the ratio m is to n is given by 2 is to 1 that is this is equal to vector a let this be equal to vector b then the position vector of r dividing the line segment joining the points p and q internally in the ratio m is to n is given by n a that is 1 into i cap plus 2 j cap minus k cap plus m b that is 2 into minus i cap plus j cap plus k cap upon m plus n and this is equal to minus i cap plus 4 j cap plus k cap upon 3 this is equal to minus 1 upon 3 i cap plus 4 upon 3 j cap plus 1 upon 3 k cap this is the position vector of the r dividing the line segment joining the points p and q internally in the ratio m is to n this completes the session hope you have understood the concept of vectors addition of vectors and multiplication of vectors