A Note From Fig . , using the triangle law, one may note that or (since which is parallelogram law. Thus, we may say that the two laws of vector addition are equivalent to each other. Properties of vector addition Property For any two vectors (Commutative property) Fig .
Proof Consider the parallelogram ABCD (Fig . ). Let then using the triangle law, from triangle ABC, we have Now, since the opposite sides of a parallelogram are equal and parallel, from Fig . , we have, .
Again using triangle law, from triangle ADC, we have Hence Property For any three vectors c (Associative property) Proof Let the vectors be represented by , respectively, as shown in Fig . (i) and (ii). Fig . Then So Fig .
– Hence Remark The associative property of vector addition enables us to write the sum of three vectors without using brackets. Note that for any vector a , we have Here, the zero vector is called the additive identity for the vector addition. . Multiplication of a Vector by a Scalar Let be a given vector and λ a scalar.
Then the product of the vector by the scalar λ , denoted as λ , is called the multiplication of vector by the scalar λ . Note that, λ is also a vector, collinear to the vector . The vector λ has the direction same (or opposite) to that of vector according as the value of λ is positive (or negative). Also, the magnitude of vector λ is | λ | times the magnitude of the vector , i.e., | λ | = | λ || | A geometric visualisation of multiplication of a vector by a scalar is given in Fig .
. Fig . When λ = – , then λ = – , which is a vector having magnitude equal to the magnitude of and direction opposite to that of the direction of . The vector – is called the negative (or additive inverse ) of vector and we always have + (– ) = (– ) + = Also, if