Numerical Problems Based on Expressing the Vectors in terms of Base Vectors and Rectangular Components of Vectors for Class 11 Physics

Addition ($\vec{A} + \vec{B}$):

Magnitude:

Subtraction ($\vec{A} – \vec{B}$):

Magnitude:

The resultant vector $\vec{R}$ is the sum of vectors $\vec{A}$ and $\vec{B}$:

The magnitude of the resultant vector is:

The unit vector $\hat{R}$ parallel to $\vec{R}$ is given by $\hat{R} = \frac{\vec{R}}{|\vec{R}|}$:

(Note: The original question text contained minor typos in the notation for $\vec{A}$ and $\vec{B}$, which have been corrected here based on standard vector notation.)

First, find the resultant $\vec{R}$ of $\vec{A}$ and $\vec{B}$:

Let the required vector be $\vec{C}$. We are given that adding $\vec{C}$ to $\vec{R}$ gives the unit vector along the z-axis, which is $\hat{k}$:

By definition, the magnitude of a unit vector is exactly 1.

Squaring both sides:

Let the sum of the three vectors be $\vec{S}$:

Combine the $\hat{i}$ and $\hat{j}$ components:

The magnitude of this sum is:

Let the magnitude of the force be $F$. The angle $\theta = 30^\circ$.

The horizontal component of the force ($F_x$) is given by:

Substitute the given values ($F_x = 50\text{ N}$):

The vertical component of the force ($F_y$) is given by: