Subtraction of a vector from another vector
The subtraction of one vector from the other is a special case of addition of two vectors in which one vector is compounded with the negative of another.
\(
\overline a - \overline b = \overline a + \left( { - \overline b } \right)
\)
Note: If \(
\overrightarrow A = A_x \hat i + A_y \hat j + A_z \hat k
\) and \(
\overrightarrow B = B_x \hat i + B_y \hat j + B_z \hat k
\)
then \(
\overrightarrow A + \overrightarrow B = \left( {A_x + B_x } \right)\hat i + \left( {A_y + B_y } \right)\hat j + \left( {A_z + B_z } \right)\hat k
\)
and \(
\overrightarrow A - \overrightarrow B = \left( {A_x - B_x } \right)\hat i + \left( {A_y - B_y } \right)\hat j + \left( {A_z - B_z } \right)\hat k
\)
Displacement
Displacement is a shortest distance between two points. It is a vector quantity.
Displacement vector
The position of the point Q with reference to the origin is represented by the position vector \(
\mathop {r_2 }\limits^ \to
\). Let the coordinates of the point Q are (x2, y2)
Similarly \(
\mathop {OP}\limits^ \to
\) represent by a position vector \(
\mathop {r_1 }\limits^ \to
\) , let the coordinates of the point P are (x1, y1)
As the displacement vector is the difference of two position vectors
\(
\mathop {r_1 }\limits^ \to
\) = x1 + y1 and \(
\mathop {r_2 }\limits^ \to
\) = x2 + y2, where \(\hat i\), \(\hat j\) are unit vectors along X, Y axis respectively.
Thus, the displacement vector \(
\Delta \mathop r\limits^ \to = \text{ }\mathop {r_2 }\limits^ \to - \mathop {r_1 }\limits^ \to
\) = (x2 – x1)\(\hat i\) + (y2 – y1)\(\hat j\)