Let's break down the problem step by step: --- ## **Given Data** - **Force:** \(\vec{F} = 14\hat{i} + 10\hat{j} - 14\hat{k}\) N - **Coordinates:** - \(O = (, , )\) - \(A = (4, 2, )\) [from the diagram] - **Angle:** The pole \(OA\) is at \(60^\circ\) from the \(x\)-axis in the \(xy\)-plane. --- # **Step 1: Find the Vector Along the Axis of the Pole (\(\vec{OA}\))** \[ \vec{OA} = (4-)\hat{i} + (2-)\hat{j} + (-)\hat{k} = 4\hat{i} + 2\hat{j} \] ### **Magnitude of OA:** \[ |\vec{OA}| = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \] ### **Unit Vector Along OA (\(\hat{u}_{OA}\)):** \[ \hat{u}_{OA} = \frac{\vec{OA}}{|\vec{OA}|} = \frac{4\hat{i} + 2\hat{j}}{2\sqrt{5}} = \frac{2}{\sqrt{5}}\hat{i} + \frac{1}{\sqrt{5}}\hat{j} \] --- # **Step 2: Calculate the Projection of \(\vec{F}\) Along the Pole** The magnitude of the projection is: \[ \text{Proj}_{OA}(\vec{F}) = \vec{F} \cdot \hat{u}_{OA} \] \[ = (14\hat{i} + 10\hat{j} - 14\hat{k}) \cdot \left(\frac{2}{\sqrt{5}}\hat{i} + \frac{1}{\sqrt{5}}\hat{j}\right) \] \[ = 14 \cdot \frac{2}{\sqrt{5}} + 10 \cdot \frac{1}{\sqrt{5}} + (-14) \cdot \] \[ = \frac{28}{\sqrt{5}} + \frac{10}{\sqrt{5}} \] \[ = \frac{38}{\sqrt{5}} \] \[ \approx \frac{38}{2.236} \approx 17.~\text{N} \] --- # **Step 3: Calculate the Magnitude of the Projection Perpendicular to the Axis of the Pole** This is found using the Pythagorean theorem: \[ |\text{Proj}_\perp| = \sqrt{|\vec{F}|^2 - |\text{Proj}_{OA}(\vec{F})|^2} \] ### **Magnitude of F:** \[ |\vec{F}| = \sqrt{14^2 + 10^2 + (-14)^2} = \sqrt{196 + 100 + 196} = \sqrt{492} \approx 22.2~\text{N} \] ### **Plug in the values:** \[ |\text{Proj}_\perp| = \sqrt{(22.2)^2 - (17.)^2} = \sqrt{492 - 289} = \sqrt{203} \approx 14.2~\text{N} \] --- # **# Final Answers** ## **1. Magnitude of the projection of \(\vec{F}\) along the axis of the pole:** \[ \boxed{17.~\text{N}} \] ## **2. Magnitude of the projection of \(\vec{F}\) perpendicular to the axis of the pole:** \[ \boxed{14.2~\text{N}} \] --- If you need the vector form of the perpendicular projection, let me know!