Displacement and Displacement vector
Displacement is a shortest distance between two points. It is a vector quantity.
The position of the point Q with reference to the origin is represented by the position vector \( \mathop {r_2 }\limits^ \to \)\( \mathop {r_2 }\limits^ \to \). Let the coordinates of the point Q are (x2, y2)
Similarly \( \mathop {OP }\limits^ \to \) represent by a position vector \( \mathop {r_1 }\limits^ \to \)\( \mathop {r_1 }\limits^ \to \), let the coordinates of the point P are (x1, y1)
As the displacement vector is the difference of two position vectors \( \mathop {r_1 }\limits^ \to \) \( \mathop {r_1 }\limits^ \to \)= x1\( \mathop i\limits^ \wedge \)\(\mathop i\limits^ \wedge \) + y1\( \mathop j\limits^ \wedge \)\(\mathop j\limits^ \wedge \) and \( \mathop {r_2 }\limits^ \to \)\( \mathop {r_2 }\limits^ \to \)= x2\( \mathop i\limits^ \wedge \) \(\mathop i\limits^ \wedge \)+ y2\(\mathop j\limits^ \wedge \), where \( \mathop i\limits^ \wedge \),\(\mathop i\limits^ \wedge ,\mathop j\limits^ \wedge \) \( \mathop j\limits^ \wedge \) are unit vectors along X, Y axis respectively.
Thus, the displacement vector \( \Delta \mathop r\limits^ \to = \text{ }\mathop {r_2 }\limits^ \to - \mathop {r_1 }\limits^ \to \)\(\Delta \mathop r\limits^ \to = {\text{ }}\mathop {{r_2}}\limits^ \to - \mathop {{r_1}}\limits^ \to \) = (x2 \(-\)– x1)\( \mathop i\limits^ \wedge \) \(\mathop i\limits^ \wedge \)+ (y2 – y1)\(\mathop j\limits^ \wedge \)
Multiplication of a vector with a scalar :
a) When a vector is multiplied by a scalar, the resultant is also a vector.
b)\(\overrightarrow A = s\overrightarrow B \), If s is positive scalar then direction of \(\overrightarrow A \) is the same as that of \(\overrightarrow B \) .
c) If s is negative then direction of \(\overrightarrow A \) is opposite to that of \(\overrightarrow B \)
d) Vector multiplication obeys commutative law when multiplied by a scalar.
i.e.,\(s\,\overrightarrow A = \overrightarrow A \,s\) where s is a scalar
e) Vector multiplication obeys associative law when multiplied by scalar.
i.e. \(m\left( {n\overrightarrow A } \right) = mn\,\overrightarrow A \) (m, n are scalars)
f) Vector multiplication obeys distributive law when multiplied by a scalar.
i.e. \(s\left( {\overrightarrow A + \overrightarrow B } \right) = s\overrightarrow A + s\overrightarrow B \) (s is a scalar).
Note: 1) A vector can be added to another vector or can be subtracted from another vector, the resultant also a vector
2) A vector can be multiplied with a scalar (or) with reciprocal of scalar, the result also vector.
3) Vector, vector division is not possible.
4) If a vector is represented by \(x\hat i + y\hat j + z\hat k\). Its length, in X – Y plane is \(\sqrt {{x^2} + {y^2}} \) , in Y – Z plane is \(\sqrt {{y^2} + {z^2}} \), in X – Z plane is \(\sqrt {{x^2} + {z^2}} \)