Today is our topic of discussion Subtraction of vectors .
Subtraction of vectors
The subtraction of the vectors u and v is u – v and it is equivalent to the addition of u and -v (opposite vector of v) i.e. u + (- v)
Triangle law of subtraction of vectors:
If the initial points of u and v are same then initial point of u – v will be the same as the final point of v and the final point of u – v will be the same as the final point of u. In short, the difference of two vectors with same initial point is the opposite vector formed by the initial points. Therefore, if u = AB , v= AC then u – v = CB , i.e. AB – AC = CB We shall prove it now.
Proof:
Line segment CA is produced such that AE = CA parallelogram AEFB is drawn. According to the parallelogram law of additions of vectors, AE + AB = vec AF.
Again AFBC is a parallelogram, as BF = AE = CA and since BF ||AE then
BF CA.
AF = CB (by transfer of vector), but AE = – v and AB = u
Therefore u – v = CB is proved.
Zero Vector
A vector whose absolute value is zero and whose direction cannot be determined is called a zero vector.
If u is any vector, then what is the value of u + (- u) ?
Let u= AB then – u = BA .
Hence, u – u = AB + BA = AA [by triangle law]
But what kind of vector is AA ? Its initial point and final point are same. Hence its length is zero. i.e. AA is to be understood as the point A. This kind of vector (whose length is zero) is called zero-vector and denoted by the symbol 0. This is the only vector which has no fixed direction and support line.
For the introduction of zero-vector we can say that u + (- u) = 0 and u + 0 = 0 + u = u Virtually trial identity is involved with zero vector.
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