The document Addition and Multiplication of a Vector by a Scalar JEE Notes | EduRev is a part of the JEE Course Mathematics (Maths) Class 12.

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**Addition of Vectors**

Let **a** and **b** be any two vectors. From the terminal point of a, vector b is drawn. Then, the vector from the initial point O of a to the terminal point B of b is called the sum of vectors a and b and is denoted by **a + b**. This is called the triangle law of addition of vectors.

**Parallelogram Law**

Let a and b be any two vectors. From the initial point of a, vector b is drawn and parallelogram OACB is completed with OA and OB as adjacent sides. The vector OC is defined as the sum of a and b. This is called the parallelogram law of addition of vectors.

The sum of two vectors is also called their resultant and the process of addition as composition.

**Properties of Vector Addition**

(i) a + b = b + a (commutativity)

(ii) a + (b + c)= (a + b)+ c (associativity)

(iii) a+ O = a (additive identity)

(iv) a + (â€” a) = 0 (additive inverse)

(v) (k_{1} + k_{2}) a = k_{1} a + k_{2}a (multiplication by scalars)

(vi) k(a + b) = k a + k b (multiplication by scalars)

(vii) |a+ b| â‰¤ |a| + |b| and |a â€“ b| â‰¥ |a| â€“ |b|

**Difference** (Subtraction) **of Vectors**

If a and b be any two vectors, then their difference a â€“ b is defined as a + (- b).

**Multiplication of a Vector by a Scalar**

Let a be a given vector and Î» be a scalar. Then, the product of the vector a by the scalar Î» is Î» a and is called the multiplication of vector by the scalar.

**Important Properties**

(i) |Î» a| = |Î»| |a|

(ii) Î» O = O

(iii) m (-a) = â€“ ma = â€“ (m a)

(iv) (-m) (-a) = m a

(v) m (n a) = mn a = n(m a)

(vi) (m + n)a = m a+ n a

(vii) m (a+b) = m a + m b

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