Physics > Basic Vectors > 8.0 Cartesian co-ordinate system
Basic Vectors
1.0 Introduction
2.0 Representation of vector
3.0 Basic definition related with vectors
3.1 Unit vector
3.2 Negative of a vector
3.3 Modulus of a vector
3.4 Multiplication and division of vectors by scalars
4.0 Types of vectors
4.1 Equal vectors
4.2 Parallel vectors
4.3 Anti-parallel vectors
4.4 Collinear vectors
4.5 Coplanar vectors
4.6 Zero or null vectors
5.0 Angle between the vectors
6.0 Addition of vectors
6.1 Triangle law of vector addition
6.2 Parallelogram law of vector addition
6.3 Relation between triangle and parallelogram law of vector addition
7.0 Subtraction of vectors
8.0 Cartesian co-ordinate system
8.1 Unit vector in cartesian co-ordinate system
8.2 Position vector of a point
8.3 Displacement vector
9.0 Resolving vector into its components
10.0 Dot product of two vectors
10.1 Properties of dot product
10.2 Condition when two vectors are perpendicular
10.3 Angle between two vectors
10.4 Geometrical meaning of scalar product
10.5 Application of dot product
11.0 Cross product of two vectors
8.3 Displacement vector
3.2 Negative of a vector
3.3 Modulus of a vector
3.4 Multiplication and division of vectors by scalars
4.2 Parallel vectors
4.3 Anti-parallel vectors
4.4 Collinear vectors
4.5 Coplanar vectors
4.6 Zero or null vectors
6.2 Parallelogram law of vector addition
6.3 Relation between triangle and parallelogram law of vector addition
8.2 Position vector of a point
8.3 Displacement vector
10.2 Condition when two vectors are perpendicular
10.3 Angle between two vectors
10.4 Geometrical meaning of scalar product
10.5 Application of dot product
Consider two points $A$ and $B$ with coordinate $\left( {a_1,b_1,c_1} \right)$ and $\left( {a_2,b_2,c_2} \right)$ respectively.
Now the particle moves from point $A$ to $B$.
So, $\overrightarrow {AB} $ will be the displacement vector as shown in the figure.
In $\Delta OAB$, from triangle law of vector addition we can write,
$$\overrightarrow {OA} + \overrightarrow {AB} = \overrightarrow {OB} $$$$\overrightarrow {AB} = \overrightarrow {OB} - \overrightarrow {OA} $$
We can write,
$$\overrightarrow {OA} = {a_1}\widehat i + {b_1}\widehat j + {c_1}\widehat k$$$$\overrightarrow {OB} = {a_2}\widehat i + {b_2}\widehat j + {c_2}\widehat k$$ As, $$\overrightarrow {AB} = \overrightarrow {OB} - \overrightarrow {OA} $$ $$\overrightarrow {AB} = \left( {{a_2} - {a_1}} \right)\widehat i + \left( {{b_2} - {b_1}} \right)\widehat j + \left( {{c_2} - {c_1}} \right)\widehat k$$
Similarly, $$\overrightarrow {BA} = \overrightarrow {OA} - \overrightarrow {OB} $$
Distance between point $A$ and $B$ is given by, $$\left| {\overrightarrow {AB} } \right| = \sqrt {{{\left( {{a_2} - {a_1}} \right)}^2} + {{\left( {{b_2} - {b_1}} \right)}^2} + {{\left( {{c_2} - {c_1}} \right)}^2}} $$
