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 Subtraction of Vectors

The negative of a vector is another vector of equal magnitude but opposite direction: e.g.  

(Diagram of two vectors pointing in opposite directions)

The difference between two vectors

The difference between two vectors is obtained by adding to the first the negative (or opposite) of the second.
( v = v_1 - v_2 = (v_1 + (-v_2)) ).
Note that ( v_2 - v_1 = -v ); if the velocities are subtracted in the reverse order, the opposite vector results. Vector subtraction is anti-commutative. The magnitude of the difference is
( D = sqrt{v_1^2 + v_2^2 + 2v_1v_2 cos(pi - \theta)} = sqrt{v_1^2 + v_2^2 - 2v_1v_2 cos \theta} ).
NB: The magnitude of a vector quantity is basically its length.

Component of a Vector

The component of a vector is its effective value in any given direction. For example, the horizontal component of a vector is its effective value in a horizontal direction. A vector may be considered as the resultant of two or more component vectors. It’s customary and most useful to resolve a vector into components along mutually perpendicular directions.
From the figure below, we see that ( v = v_x + v_y ). But ( v_x = v cos alpha ), ( v_y = v sin alpha ). Defining unit vector ( v_x ) and ( v_y ) in the direction of the X and Y-axis, we note that:
( v_x = OA = u_x v_x, quad v_y = OB = u_y v_y ).
Therefore ( v = u_x v_x + u_y v_y ).
In three dimensions, we have ( v = u_x v_x + u_y v_y + u_z v_z ).

Multiplication of Vectors

Operations of addition and subtraction can be carried out among like vectors. However, in the case of vector multiplication, vectors of different kinds representing different physical quantities can be multiplied, giving rise to another meaningful physical quantity. For example,
( mathbf{F}_B = q_0 mathbf{v} \times mathbf{B} ),
where ( mathbf{F}_B ) is the magnetic deflecting force in the magnetic field, ( mathbf{v} ) is the drift velocity, and ( mathbf{B} ) is the magnetic inductance.

There are three kinds of operations for vector multiplication:
(i) Vector × Scalar = Vector
(ii) Vector × Vector = Scalar
(iii) Vector × Vector = Vector

Multiplication of a vector with a scalar

If a vector ( a ) is multiplied with an arbitrary number ( n ) (a scalar ( n )), the resultant vector ( R ) will be ( n ) times the magnitude of ( a ) but the direction of ( R ) remains the same.
( n \times a = R = na )
Hence multiplication of a vector and a scalar gives a vector quantity in the same direction.