Study Notes
Addition, Subtraction and Resolution of Vectors
Addition of Vectors
Let two vectors $\vec{A}$ and $\vec{B}$ be inclined at an angle $\theta$ to each other.
If we add them, we get a resultant vector $\vec{R}$ which makes an angle $\alpha$ with vector $\vec{A}$.
Magnitude of Resultant Vector
$$ R = \sqrt{A^2 + B^2 + 2AB\cos\theta} $$
Direction of Resultant Vector
The angle $\alpha$ made by resultant $\vec{R}$ with vector $\vec{A}$ is:
$$ \tan\alpha = \frac{B\sin\theta}{A + B\cos\theta} $$
Important Cases (Very Important for Exams)
1. When $\theta = 0^\circ$ (same direction)
$$ R = A + B,\qquad \alpha = 0^\circ $$
2. When $\theta = 90^\circ$
$$ R = \sqrt{A^2 + B^2} $$
$$ \alpha = \tan^{-1}\left(\frac{B}{A}\right) $$
3. When $\theta = 180^\circ$ (opposite direction)
$$ R = |A - B| $$
- If $A > B$, then $\alpha = 0^\circ$
- If $B > A$, then $\alpha = 180^\circ$
4. When $A = B$
$$ R = 2A\cos\frac{\theta}{2} $$
Limit of Resultant Vector
$$ |A - B| \le |\vec{A} + \vec{B}| \le A + B $$
This means the resultant always lies between the difference and the sum of the two vectors.
Difference of Vectors
The difference of two vectors $\vec{A}$ and $\vec{B}$ is:
$$ \vec{R} = \vec{A} - \vec{B} $$
The resultant $\vec{R}$ makes an angle $\alpha$ with vector $\vec{A}$.
Magnitude of Resultant (Difference)
$$ R = \sqrt{A^2 + B^2 - 2AB\cos\theta} $$
Direction of Resultant
$$ \tan\alpha = \frac{B\sin\theta}{A - B\cos\theta} $$
Important Cases
1. When $\theta = 0^\circ$
$$ R = A - B $$
- If $A > B$, then $\alpha = 0^\circ$
- If $B > A$, then $\alpha = 180^\circ$
2. When $\theta = 90^\circ$
$$ R = \sqrt{A^2 + B^2} $$
$$ \alpha = \tan^{-1}\left(\frac{B}{A}\right) $$
3. When $\theta = 180^\circ$
$$ R = A + B,\qquad \alpha = 0^\circ $$
4. When $A = B$
$$ R = 2A\sin\frac{\theta}{2} $$
Limit of Vector Difference
$$ |A - B| \le |\vec{A} - \vec{B}| \le A + B $$
Resolution of Vectors
A vector can be resolved into many components.
- In a plane → 2 rectangular components
- In space → 3 rectangular components
Important points:
- Components of a vector are also vectors.
- A component of a vector is always less than or equal to the vector itself.
Rectangular Components of a Vector
For a vector:
$$
\vec{A} = A_x \hat{i} + A_y \hat{j}
$$
Horizontal Component
$$ A_x = A\cos\theta $$
Vertical Component
$$ A_y = A\sin\theta $$
If Components are Given
If $A_x$ and $A_y$ are known:
Magnitude of Vector
$$ A = \sqrt{A_x^2 + A_y^2} $$
Direction (angle with horizontal)
$$ \alpha = \tan^{-1}\left(\frac{A_y}{A_x}\right) $$