find the angle between the vectors $\langle -9, 7 \
angle$ and $\langle 9, 6 \
angle$. carry your intermediate computations to at least 4 decimal places. round your final answer to the nearest degree.

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Vector Magnitude", "Vector Dot Product", "Angle Between Vectors" ], "new_concepts": [], "current_concepts": [ "Vector Magnitude", "Vector Dot Product", "Angle Between Vectors" ] } ## Calculate the dot product of the vectors Using the *Vector Dot Product* knowledge point $$ \begin{aligned} \mathbf{u} \cdot \mathbf{v} &= (-9)(9) + (7)(6) \ &= -81 + 42 \ &= -39 \end{aligned} $$ ## Calculate the magnitudes of both vectors Using the *Vector Magnitude* knowledge point $$ \begin{aligned} \|\mathbf{u}\| &= \sqrt{(-9)^2 + 7^2} = \sqrt{81 + 49} = \sqrt{130} \approx 11.4018 \ \|\mathbf{v}\| &= \sqrt{9^2 + 6^2} = \sqrt{81 + 36} = \sqrt{117} \approx 10.8167 \end{aligned} $$ ## Calculate the cosine of the angle Using the *Angle Between Vectors* knowledge point $$ \begin{aligned} \cos(\theta) &= \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|} \ &= \frac{-39}{\sqrt{130} \cdot \sqrt{117}} \ &= \frac{-39}{\sqrt{15210}} \ &\approx \frac{-39}{123.3288} \ &\approx -0.3162 \end{aligned} $$ ## Find the angle in degrees Using the *Angle Between Vectors* knowledge point $$ \begin{aligned} \theta &= \arccos(-0.3162277) \ &\approx 108.4349^\circ \ &\approx 108^\circ \end{aligned} $$ Find the angle between the vectors $\langle -9, 7 angle$ and $\langle 9, 6 angle$. Carry your intermediate computations to at least 4 decimal places. Round your final answer to the nearest degree. 108$^\circ$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Angle Between Vectors" ] }

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Question

Explanation:

Calculate the dot product of the vectors

Calculate the magnitudes of both vectors

Calculate the cosine of the angle

Find the angle in degrees

Answer: