a shopper pushes a grocery cart with a force of $45\text{ n}$ downward and to the right, as shown in the figure below. the angle the handle makes with the horizontal is $42^{\circ}$. 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 shopper pushes a grocery cart with a force of $45\text{ n}$ downward and to the right, as shown in the figure below. the angle the handle makes with the horizontal is $42^{\circ}$. 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 parameters Using the *Vector Components* knowledge point $$ \begin{aligned} &\|\mathbf{F}\| = 45\text{ N}\ &\theta = -42^\circ \quad (\text{or } 318^\circ) \end{aligned} $$ ## Calculate the components Using the *Trigonometric Resolution* knowledge point $$ \begin{aligned} &a = 45 \cos(-42^\circ) \approx 33.44\ &b = 45 \sin(-42^\circ) \approx -30.11 \end{aligned} $$ ## Write the vector form Using the *Vector Components* knowledge point $$ \mathbf{F} = 33.44\mathbf{i} - 30.11\mathbf{j} $$ $\mathbf{F} = 33.44\mathbf{i} - 30.11\mathbf{j}$ { "elements": [ { "type": "point", "params": [[0, 0]], "properties": { "name": "O", "size": 3, "color": "#583C87", "withLabel": false } }, { "type": "line", "params": [[0, 0], [5, 0]], "properties": { "strokeColor": "gray", "strokeWidth": 1, "dash": 2 } }, { "type": "arrow", "params": [[0, 0], [3.34, -3.01]], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3, "name": "F", "withLabel": true, "labelPosition": [1.8, -1.5] } }, { "type": "angle", "params": [[3.34, -3.01], [0, 0], [5, 0]], "properties": { "radius": 1.2, "strokeColor": "#F2557F", "fillColor": "rgba(242, 85, 127, 0.2)", "name": "42°", "withLabel": true } } ] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Vector Components" ] }

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Question

Explanation:

Identify the vector parameters

Calculate the components

Write the vector form

Answer: