To solve the problem of finding the resultant force when two forces of 10 N each act at right angles to each other (one towards the north and the other towards the east), we can use vector addition. Let's break this down step-by-step. ### Step 1: Understanding the Forces We have two forces: - **Force 1 (F1)**: 10 N directed towards the North. - **Force 2 (F2)**: 10 N directed towards the East. ### Step 2: Representing the Forces as Vectors In physics, we can represent forces as vectors. We can denote: - **F1** as \( \vec{F_1} = 10 \, \text{N} \, \hat{j} \) (where \( \hat{j} \) represents the unit vector in the north direction). - **F2** as \( \vec{F_2} = 10 \, \text{N} \, \hat{i} \) (where \( \hat{i} \) represents the unit vector in the east direction). ### Step 3: Calculating the Resultant Force To find the resultant force \( \vec{R} \), we can use the Pythagorean theorem since the forces are perpendicular to each other: \[ R = \sqrt{F_1^2 + F_2^2} \] Substituting the values: \[ R = \sqrt{(10 \, \text{N})^2 + (10 \, \text{N})^2} \] \[ R = \sqrt{100 + 100} \] \[ R = \sqrt{200} \] \[ R = 10\sqrt{2} \, \text{N} \] ### Step 4: Determining the Direction of the Resultant Force To find the direction of the resultant force, we can use trigonometry. The angle \( \theta \) that the resultant makes with the east direction can be found using the tangent function: \[ \tan(\theta) = \frac{F_1}{F_2} = \frac{10 \, \text{N}}{10 \, \text{N}} = 1 \] Thus, \[ \theta = \tan^{-1}(1) = 45^\circ \] Since the force towards the north is the vertical component and the force towards the east is the horizontal component, the angle is measured from the east towards the north, which means the direction of the resultant force is \( 45^\circ \) towards the north of east. ### Final Result The magnitude of the resultant force is \( 10\sqrt{2} \, \text{N} \) and the direction is \( 45^\circ \) towards the north of east. ### Answer The correct option is **D. 10√2N, 45°E**. ### Explanation of Other Options - **Option A (20N, 45°E)**: This option incorrectly adds the magnitudes of the two forces directly. Since they are at right angles, we cannot simply add them; we must use vector addition. - **Option B (20N, 45°W)**: Similar to Option A, this option incorrectly adds the magnitudes and also misrepresents the direction. The resultant is not 20 N, and the angle is not towards the west. - **Option C (10√2N, 45°W)**: This option correctly states the magnitude but incorrectly states the direction. The resultant is directed towards the east, not the west. ### Revision Summary - The resultant force of two perpendicular forces can be calculated using the Pythagorean theorem. - The direction of the resultant can be found using trigonometric functions (tangent). - The correct resultant magnitude is \( 10\sqrt{2} \, \text{N} \) and the direction is \( 45^\circ \) towards the east. - Always remember to consider the direction of forces when determining the resultant vector.