Multiplication of a Vector by a Scalar




Let `vec a` be a given vector and `lambda` a scalar. Then the product of the vector `vec a` by the scalar `lambda` , denoted as `lambda vec a` , is called the  multiplication of vector `vec a` by the scalar `lambda`. `lambda vec a` is also a vector, collinear to the vector `vec a` .  The vector `lambda vec a`  has the direction same (or opposite) to that of vector `vec a` according as the value of λ is positive (or negative). Also, the magnitude of vector λ`vec a` is |λ| times the magnitude of the vector `vec a` , i.e.,
 |λ `vec a`| = |λ| |`vec a`| 
A geometric visualisation of multiplication of a vector by a scalar is given in fig.

When λ = –1, then λ`vec a` = – `vec a`, which is a vector having magnitude equal to the magnitude of  and direction opposite to that of the direction of `vec a` . The vector –`vec a` is called the negative (or additive inverse) of vector `vec a` and we always have
`vec a + (-vec a) = (-vec a) + vec a = 0`
Also, if  λ = `1/|vec a|`  ,  provided `vec a` ≠ 0 i.e.  is not a null vector, then
|λ `veca`| =`|λ| |vec a| = 1/|vec a||vec a| = 1`
So, λ `vec a` represents the unit vector in the direction of `vec a`. We write it as
`hat a = 1/|vec a| vec a`
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