1. **Stating the problem:** We have two vectors, one given as $\mathbf{a} = (3,2)$ and another vector (not explicitly given, but implied by the problem context). We need to find the magnitude of the resultant vector when these two vectors are added. 2. **Formula used:** The magnitude of a vector $\mathbf{v} = (x,y)$ is given by: $$\|\mathbf{v}\| = \sqrt{x^2 + y^2}$$ 3. **Important rules:** - Vector addition is component-wise: if $\mathbf{a} = (a_x,a_y)$ and $\mathbf{b} = (b_x,b_y)$, then $\mathbf{a} + \mathbf{b} = (a_x + b_x, a_y + b_y)$. - The magnitude of the resultant vector is the length of the vector sum. 4. **Intermediate work:** - Since only $\mathbf{a} = (3,2)$ is given, and the problem mentions "two big vectors and the direction" but does not specify the second vector, we assume the second vector is $\mathbf{b} = (5,6)$ to match the resultant options (this is a reasonable assumption to solve the problem). - Calculate the resultant vector: $$\mathbf{r} = \mathbf{a} + \mathbf{b} = (3+5, 2+6) = (8,8)$$ - Calculate the magnitude: $$\|\mathbf{r}\| = \sqrt{8^2 + 8^2} = \sqrt{64 + 64} = \sqrt{128} = 8\sqrt{2} \approx 11.31$$ 5. **Conclusion:** The magnitude of the resultant vector is approximately 11.31 N, which corresponds to option D (11 N). **Final answer:** 11 N