suppose the vector $\mathbf{v}$ has magnitude 23 and makes an angle of $153^\circ$ with the positive $x$-axis (measured counterclockwise), when $\mathbf{v}$ is in standard position.

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

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

Question

suppose the vector $\mathbf{v}$ has magnitude 23 and makes an angle of $153^\circ$ with the positive $x$-axis (measured counterclockwise), when $\mathbf{v}$ is in standard position.

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

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", "blank_filling_formatter", "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 given parameters Using the *Vector Components* knowledge point $$ \begin{aligned} &\|\mathbf{v}\| = 23\ &\theta = 153^\circ \end{aligned} $$ ## Calculate the horizontal component Using the *Trigonometric Resolution* knowledge point $$ \begin{aligned} &a = \|\mathbf{v}\| \cos(\theta)\ &a = 23 \cos(153^\circ)\ &a \approx -20.4928\ &a \approx -20.49 \end{aligned} $$ ## Calculate the vertical component Using the *Trigonometric Resolution* knowledge point $$ \begin{aligned} &b = \|\mathbf{v}\| \sin(\theta)\ &b = 23 \sin(153^\circ)\ &b \approx 10.4417\ &b \approx 10.44 \end{aligned} $$ ## Express the vector in standard form Using the *Vector Components* knowledge point $$ \begin{aligned} &\mathbf{v} = a\mathbf{i} + b\mathbf{j}\ &\mathbf{v} = -20.49\mathbf{i} + 10.44\mathbf{j} \end{aligned} $$ Suppose the vector $\mathbf{v}$ has magnitude 23 and makes an angle of $153^\circ$ with the positive $x$-axis (measured counterclockwise), when $\mathbf{v}$ is in standard position. Write $\mathbf{v}$ in the form $\mathbf{v} = a\mathbf{i} + b\mathbf{j}$. Do not round any intermediate computations, and round the values in your answer to the nearest hundredth. $\mathbf{v} =$ $-20.49\mathbf{i} + 10.44\mathbf{j}$ { "elements": [ { "type": "point", "params": [[0, 0]], "properties": { "name": "O", "size": 3, "color": "#583C87", "withLabel": false } }, { "type": "point", "params": [[-2.049, 1.044]], "properties": { "name": "v", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "arrow", "params": [[0, 0], [-2.049, 1.044]], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3 } }, { "type": "line", "params": [[0, 0], [1, 0]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 1, "dash": 2 } }, { "type": "angle", "params": [[1, 0], [0, 0], [-2.049, 1.044]], "properties": { "radius": 0.8, "fillColor": "rgba(140, 85, 242, 0.2)", "strokeColor": "#8C55F2", "withLabel": true, "name": "153°" } } ] } { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Vector Components" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the given parameters

Calculate the horizontal component

Calculate the vertical component

Express the vector in standard form

Answer: