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Math Problem: Let A=(1,−11,2),B=(3,−4,1)A=(1,-11,2), B=(3,-4,1), and C=(−2,1,−1)C=(-2,1,-1). Compute $\angle ...
Mar 29, 2024
Math Problem: Let A=(1,−11,2),B=(3,−4,1)A=(1,-11,2), B=(3,-4,1), and C=(−2,1,−1)C=(-2,1,-1). Compute ∠ABC\angle A B C, in degrees.
Solution by Steps
step 1
To find the angle ∠ABC \angle ABC , we first need to find the vectors AB→ \overrightarrow{AB} and BC→ \overrightarrow{BC}
step 2
The vector AB→ \overrightarrow{AB} is found by subtracting coordinates of A A from B B : AB→=B−A=(3−1,−4−(−11),1−2) \overrightarrow{AB} = B - A = (3 - 1, -4 - (-11), 1 - 2)
step 3
Simplifying the coordinates of AB→ \overrightarrow{AB} gives us AB→=(2,7,−1) \overrightarrow{AB} = (2, 7, -1)
step 4
Similarly, the vector BC→ \overrightarrow{BC} is BC→=C−B=(−2−3,1−(−4),−1−1) \overrightarrow{BC} = C - B = (-2 - 3, 1 - (-4), -1 - 1)
step 5
Simplifying the coordinates of BC→ \overrightarrow{BC} gives us BC→=(−5,5,−2) \overrightarrow{BC} = (-5, 5, -2)
step 6
The angle between two vectors u→ \overrightarrow{u} and v→ \overrightarrow{v} can be found using the dot product formula: cos⁡(θ)=u→⋅v→∣u→∣∣v→∣ \cos(\theta) = \frac{\overrightarrow{u} \cdot \overrightarrow{v}}{|\overrightarrow{u}| |\overrightarrow{v}|}
step 7
Calculate the dot product AB→⋅BC→=(2)(−5)+(7)(5)+(−1)(−2) \overrightarrow{AB} \cdot \overrightarrow{BC} = (2)(-5) + (7)(5) + (-1)(-2)
step 8
Simplifying the dot product gives us AB→⋅BC→=−10+35+2=27 \overrightarrow{AB} \cdot \overrightarrow{BC} = -10 + 35 + 2 = 27
step 9
Calculate the magnitudes ∣AB→∣=22+72+(−1)2 |\overrightarrow{AB}| = \sqrt{2^2 + 7^2 + (-1)^2} and ∣BC→∣=(−5)2+52+(−2)2 |\overrightarrow{BC}| = \sqrt{(-5)^2 + 5^2 + (-2)^2}
step 10
Simplifying the magnitudes gives us ∣AB→∣=4+49+1=54 |\overrightarrow{AB}| = \sqrt{4 + 49 + 1} = \sqrt{54} and ∣BC→∣=25+25+4=54 |\overrightarrow{BC}| = \sqrt{25 + 25 + 4} = \sqrt{54}
step 11
Now we can find cos⁡(θ)=275454 \cos(\theta) = \frac{27}{\sqrt{54} \sqrt{54}}
step 12
Simplifying cos⁡(θ) \cos(\theta) gives us cos⁡(θ)=2754=12 \cos(\theta) = \frac{27}{54} = \frac{1}{2}
step 13
To find the angle in degrees, we use the inverse cosine function: θ=cos⁡−1(12) \theta = \cos^{-1}\left(\frac{1}{2}\right)
step 14
The value of cos⁡−1(12) \cos^{-1}\left(\frac{1}{2}\right) is 60∘ 60^\circ
Answer
∠ABC=60∘ \angle ABC = 60^\circ
Key Concept
Angle between two vectors using dot product
Explanation
The angle between two vectors is calculated by taking the inverse cosine of the dot product of the vectors divided by the product of their magnitudes.
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