0Vector is that quantity that has direction and magnitude
for example force, velocity, acceleration and displacement
The scaler is that quantity which has the only magnitude
for example speed, volume, mass temperature, power, energy and time
The measure of a vector is a scalar quantity.
A scalar can multiply vectors. The result is another vector.
Suppose c is a scalar and v = (a, b) is a vector, then the scalar multiplication would be cv = c (a, b) = (ca, cb). Hence each component of vector gets multiplied by the scalar.
If two vectors are of the same dimension, they can be added or subtracted from each other. The result is gain a vector.
Some Basic Rules of Algebra of Vectors:
a.a = |a|2 = a2
a.b = b.a
a.0 = 0
a.b = (a cos q)b = (projection of a on b)b = (projection of b on a) a
a.(b + c) = a.b + a.c (This is also termed as the distributive law)
(la).(mb) = lm (a.b)
(a ± b)2 = (a ± b) . (a ± b) = a2 + b2 ± 2a.b If a and b are non-zero, then the angle between them is given by cos θ = a.b/|a||b|
a x a = 0
a x b = - (b x a)
a x (b + c) = a x b + a x c
