Multiplication of a vector with a scalar :
a) When a vector is multiplied by a scalar, the resultant is also a vector.
b) \(
\overrightarrow A = s\overrightarrow B
\), If s is positive scalar then direction of \(\overrightarrow A\) is the same as that of \(\overrightarrow B\).
c) If s is negative then direction of \(\overrightarrow A\) is opposite to that of \(\overrightarrow B\)
d) Vector multiplication obeys commutative law when multiplied by a scalar.
i.e., \(
s\,\overrightarrow A = \overrightarrow A \,s
\) where s is a scalar
e) Vector multiplication obeys associative law when multiplied by scalar.
i.e. \(
m\left( {n\overrightarrow A } \right) = mn\,\overrightarrow A
\) (m, n are scalars)
f) Vector multiplication obeys distributive law when multiplied by a scalar.
i.e. \(
s\left( {\overrightarrow A + \overrightarrow B } \right) = s\overrightarrow A + s\overrightarrow B
\) (s is a scalar).
Note:
1) A vector can be added to another vector or can be subtracted from another vector, the resultant also a vector
2) A vector can be multiplied with a scalar (or) with reciprocal of scalar, the result also vector.
3) Vector, vector division is not possible.
4) If a vector is represented by \(
x\hat i + y\hat j + z\hat k
\). Its length, in X – Y plane is \(
\sqrt {x^2 + y^2 }
\) , in Y – Z plane is \(
\sqrt {y^2 + z^2 }
\) , in X – Z plane is \(
\sqrt {x^2 + z^2 }
\).