Vector Solutions, Shortcuts, Formulas
Multiplication of a Vector by a Scalar
Let be a given vector and λ a scalar. Then the product of the vector a by the scalar λ , denoted as , is called the multiplication of vector , by the scalar λ. Note that , , is also a vector, collinear to the vector , . The vector , 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 , Is λ times the magnitude of the vector , i.e ,
A geometric visualisation of multiplication of a vector by a scalar is given in( Fig 10.12)
When which is a vector having magnitude equal to the magnitude of the vector is called negative vector .(or additive inverse ) of vector a and we always have .
. Also , if , provide 0 , i.e is not a null vector , then represents the unit vector in the direction of. We write it as
Note:
 One may observe that whatever be the value of λ , the vector is always collinear to the vector . In fact, two vectors are collinear if and only if there exists a nonzero scalar λ such that . If the vectors are given in component form , i.e ,
 If two vectors are collinear if and only if
 If ,this is called the direction of ratio
 In case if it is given that l ,m ,n are direction cosine of a vector , then is the unit vector in the direction of that vector, where α , β and γ are the angles which the vector makes with x , y and z axes respectively .
Example: find the unit vector in the direction of vector
Solution: The unit vector in the direction of a vector is given by
Now
Therefore
Example: Find the unit vector in the direction of the sum of the vector
Solution: The sum of the given vector , Let
, Thus the require unit vector is
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