Let's break down the step-by-step solution for finding the Cartesian representation of the third force \(\vec{F}_3\), given: - \(\vec{F}_1\) and \(\vec{F}_2\) both have magnitudes of \(800.\,\text{N}\) - The resultant of all three forces is \(400.\,\text{N}\) in the positive \(y\)-direction (that is, \(+400.\,\hat{j}\)) - The direction angles of the forces are referenced in a diagram (not shown), so I'll assume the standard configuration for such problems: - \(\vec{F}_1\) at \(^\circ\) (along the \(+x\)-axis) - \(\vec{F}_2\) at \(90^\circ\) (along the \(+y\)-axis) - \(\vec{F}_3\) is unknown, to be determined If you have a different configuration from a diagram, let me know, but here's the solution under this standard assumption. --- ## Step 1: Write Forces in Cartesian Form ### \(\vec{F}_1\) - Magnitude: \(800.\,\text{N}\) - Direction: \(^\circ\) (along \(+x\)) - Cartesian form: \[ \vec{F}_1 = 800.\,\hat{i} \] ### \(\vec{F}_2\) - Magnitude: \(800.\,\text{N}\) - Direction: \(90^\circ\) (along \(+y\)) - Cartesian form: \[ \vec{F}_2 = 800.\,\hat{j} \] ### \(\vec{F}_3\) - Cartesian form: \[ \vec{F}_3 = a\,\hat{i} + b\,\hat{j} \] where \(a\) and \(b\) are to be determined. --- ## Step 2: Write the Vector Sum Given: \[ \vec{F}_1 + \vec{F}_2 + \vec{F}_3 = 400.\,\hat{j} \] Substitute values: \[ (800.\,\hat{i}) + (800.\,\hat{j}) + (a\,\hat{i} + b\,\hat{j}) = 400.\,\hat{j} \] Combine like terms: \[ (800. + a)\,\hat{i} + (800. + b)\,\hat{j} = 400.\,\hat{j} \] This yields two equations: \[ \begin{cases} 800. + a = \\ 800. + b = 400. \end{cases} \] --- ## Step 3: Solve for \(a\) and \(b\) ### For \(a\): \[ 800. + a = \implies a = -800. \] ### For \(b\): \[ 800. + b = 400. \implies b = 400. - 800. = -400. \] --- ## Step 4: Write the Final Answer \[ \vec{F}_3 = -800.\,\hat{i} - 400.\,\hat{j} \] Or in boldface ijk notation: \[ \boxed{\vec{F}_3 = \mathbf{-800.\,\hat{i}\ -\ 400.\,\hat{j}\ \text{N}}} \] --- ### **Summary Table** | Force | Cartesian Form | |---------|-------------------------| | \(\vec{F}_1\) | \(800.\,\hat{i}\) | | \(\vec{F}_2\) | \(800.\,\hat{j}\) | | \(\vec{F}_3\) | \(\mathbf{-800.\,\hat{i}\ -\ 400.\,\hat{j}}\) | --- If the direction angles for the forces are different in your diagram, please provide them so I can adjust the solution accordingly.