Physics > Basic Vectors > 6.0 Addition of vectors
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
6.1 Triangle law of vector addition
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 vectors $\overrightarrow A $ and $\overrightarrow B $ are represented in magnitude and direction by the two sides of a triangle taken in the same order, then the resultant is given both in magnitude and direction by the third side taken in the reverse order.
The diagram is as shown below,
Mathematically,
$$\overrightarrow R = \overrightarrow A + \overrightarrow B $$
Tail of $\overrightarrow B $ will be joined with head of $\overrightarrow A $. Then the resultant will be from tail of $\overrightarrow A $ to the head of $\overrightarrow B $.
Important derivation
We can write,
$OP = \left| {\overrightarrow A } \right| = A$
$PR = \left| {\overrightarrow B } \right| = B$
$OR = \left| {\overrightarrow R } \right| = R$
$\theta :$ Angle between $\overrightarrow A $ and $\overrightarrow B $
$\alpha :$ Angle between resultant $\left( {\overrightarrow R } \right)$ and $\overrightarrow A $
In $\Delta PQR$ we can write,
$$\cos \theta = \frac{{PQ}}{{PR}}$$$$PQ = PR\cos \theta $$ or $$PQ = B\cos \theta \quad ...(i)$$ Similarly, $$\sin \theta = \frac{{QR}}{{PR}}$$$$QR = PR\sin \theta $$or $$QR = B\sin \theta \quad ...(ii)$$
In $\Delta OQR$ we can write,
$$OQ=OP+PQ$$$$OQ=A+B\cos \theta$$$$QR=B\sin \theta$$
So, we can find $OR$ from pythagorous theorem,
$$O{R^2} = O{Q^2} + Q{R^2}$$$$O{R^2} = {\left( {A + B\cos \theta } \right)^2} + {\left( {B\sin \theta } \right)^2}$$$$O{R^2} = {A^2} + {B^2}{\cos ^2}\theta + 2AB\cos \theta + {B^2}{\sin ^2}\theta $$$$O{R^2} = {A^2} + {B^2}\left( {{{\cos }^2}\theta + {{\sin }^2}\theta } \right) + 2AB\cos \theta $$$$O{R^2} = {A^2} + {B^2} + 2AB\cos \theta $$$$OR = \sqrt {{A^2} + {B^2} + 2AB\cos \theta } $$ or $$\left| {\overrightarrow R } \right| = \sqrt {{A^2} + {B^2} + 2AB\cos \theta } $$
Also, $\alpha $ is the angle between the resultant and the vector $\overrightarrow A $.
In $\Delta OQR$ we can write,
$$\tan \alpha = \frac{{QR}}{{OQ}}$$$$\tan \alpha = \frac{{B\sin \theta }}{{A + B\cos \theta }}$$
