a worker pushes a wheelbarrow with a force of $40\text{ n}$ downward and to the left, as shown in the figure below. the angle the handle makes with the horizontal is $34^\circ$. note that the figure is not drawn to scale.

write the force vector $\mathbf{f}$ in the form $\mathbf{f} = a\mathbf{i} + b\mathbf{j}$.

do not round any intermediate computations, and round the values in your answer to the nearest hundredth.

Question

a worker pushes a wheelbarrow with a force of $40\text{ n}$ downward and to the left, as shown in the figure below. the angle the handle makes with the horizontal is $34^\circ$. note that the figure is not drawn to scale.

write the force vector $\mathbf{f}$ in the form $\mathbf{f} = a\mathbf{i} + b\mathbf{j}$.

do not round any intermediate computations, and round the values in your answer to the nearest hundredth.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Vector Components", "Trigonometric Resolution" ], "new_concepts": [], "current_concepts": [ "Vector Components", "Trigonometric Resolution" ] } ## Identify the vector magnitude and direction Using the **Vector Components** and **Trigonometric Resolution** knowledge points $$ \begin{aligned} &\|\mathbf{F}\| = 40 \ &\theta = 180^\circ + 34^\circ = 214^\circ \end{aligned} $$ ## Calculate the horizontal component Using the **Vector Components** and **Trigonometric Resolution** knowledge points $$ \begin{aligned} a &= 40 \cos(214^\circ) \ a &\approx -33.16 \end{aligned} $$ ## Calculate the vertical component Using the **Vector Components** and **Trigonometric Resolution** knowledge points $$ \begin{aligned} b &= 40 \sin(214^\circ) \ b &\approx -22.37 \end{aligned} $$ ## Formulate the final vector Using the **Vector Components** knowledge point $$ \mathbf{F} = -33.16\mathbf{i} - 22.37\mathbf{j} $$ $\mathbf{F} = -33.16\mathbf{i} - 22.37\mathbf{j}$ { "elements": [ { "type": "line", "params": [[-5, 0], [5, 0]], "properties": { "strokeColor": "#cccccc", "strokeWidth": 1, "dash": 1 } }, { "type": "line", "params": [[0, -5], [0, 5]], "properties": { "strokeColor": "#cccccc", "strokeWidth": 1, "dash": 1 } }, { "type": "arrow", "params": [[0, 0], [-3.316, -2.237]], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3, "name": "F", "withLabel": true, "labelPosition": [-2, -1.5] } }, { "type": "arrow", "params": [[0, 0], [-3.316, 0]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2, "dash": 2, "name": "a i", "withLabel": true } }, { "type": "arrow", "params": [[-3.316, 0], [-3.316, -2.237]], "properties": { "strokeColor": "#F2557F", "strokeWidth": 2, "dash": 2, "name": "b j", "withLabel": true } }, { "type": "angle", "params": [[-5, 0], [0, 0], [-3.316, -2.237]], "properties": { "radius": 1, "strokeColor": "#F28A55", "fillColor": "rgba(242, 138, 85, 0.3)", "name": "34°", "withLabel": true } } ] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Vector Components" ] }

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Question

Explanation:

Identify the vector magnitude and direction

Calculate the horizontal component

Calculate the vertical component

Formulate the final vector

Answer: