Question:
The angle between a line with direction ratios 2, 2, 1 and a line joining $(3, 1, 4)$ and $(7, 2, 12)$ is
The angle between a line with direction ratios 2, 2, 1 and a line joining $(3, 1, 4)$ and $(7, 2, 12)$ is
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Always simplify the ratios if possible to keep numbers small! Here, the magnitudes work out to clean whole integers ($3$ and $9$), making the final fraction reduction straightforward and quick on scratch paper.
Updated On: Jun 3, 2026
- $\cos^{-1}\left(\frac{1}{3}\right)$
- $\cos^{-1}\left(\frac{1}{2}\right)$
- $\cos^{-1}\left(\frac{2}{3}\right)$
- $\cos^{-1}\left(\frac{\sqrt{3}}{2}\right)$
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Question:
We need to find the angle $\theta$ between two lines in three-dimensional space. The direction ratios of the first line are explicitly provided, while the second line is defined by two points it passes through.
Step 2: Key Formula or Approach:
The angle $\theta$ between two lines with direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ is calculated using the formula: $$ \cos\theta = \frac{|a_1 a_2 + b_1 b_2 + c_1 c_2|}{\sqrt{a_1^2 + b_1^2 + c_1^2}\sqrt{a_2^2 + b_2^2 + c_2^2}} $$
Step 3: Detailed Explanation:
First, let's establish the direction ratios for both lines:
• Line 1: $(a_1, b_1, c_1) = (2, 2, 1)$
• Line 2 connects $P(3, 1, 4)$ and $Q(7, 2, 12)$. Its direction ratios are found by taking coordinate differences: $$ a_2 = 7 - 3 = 4, \quad b_2 = 2 - 1 = 1, \quad c_2 = 12 - 4 = 8 $$ So, $(a_2, b_2, c_2) = (4, 1, 8)$.
Now, let's evaluate the component parts for our angle formula:
• Numerator (dot product sum): $$ a_1 a_2 + b_1 b_2 + c_1 c_2 = (2 \times 4) + (2 \times 1) + (1 \times 8) = 8 + 2 + 8 = 18 $$
• Magnitude of Line 1 vector: $$ \sqrt{2^2 + 2^2 + 1^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3 $$
• Magnitude of Line 2 vector: $$ \sqrt{4^2 + 1^2 + 8^2} = \sqrt{16 + 1 + 64} = \sqrt{81} = 9 $$
Substitute these calculated values back into the cosine equation: $$ \cos\theta = \frac{18}{3 \times 9} = \frac{18}{27} $$ Simplifying the fraction by dividing both numbers by 9: $$ \cos\theta = \frac{2}{3} \implies \theta = \cos^{-1}\left(\frac{2}{3}\right) $$
Step 4: Final Answer:
The angle between the two lines is $\cos^{-1}\left(\frac{2}{3}\right)$, which corresponds to option (C).
We need to find the angle $\theta$ between two lines in three-dimensional space. The direction ratios of the first line are explicitly provided, while the second line is defined by two points it passes through.
Step 2: Key Formula or Approach:
The angle $\theta$ between two lines with direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ is calculated using the formula: $$ \cos\theta = \frac{|a_1 a_2 + b_1 b_2 + c_1 c_2|}{\sqrt{a_1^2 + b_1^2 + c_1^2}\sqrt{a_2^2 + b_2^2 + c_2^2}} $$
Step 3: Detailed Explanation:
First, let's establish the direction ratios for both lines:
• Line 1: $(a_1, b_1, c_1) = (2, 2, 1)$
• Line 2 connects $P(3, 1, 4)$ and $Q(7, 2, 12)$. Its direction ratios are found by taking coordinate differences: $$ a_2 = 7 - 3 = 4, \quad b_2 = 2 - 1 = 1, \quad c_2 = 12 - 4 = 8 $$ So, $(a_2, b_2, c_2) = (4, 1, 8)$.
Now, let's evaluate the component parts for our angle formula:
• Numerator (dot product sum): $$ a_1 a_2 + b_1 b_2 + c_1 c_2 = (2 \times 4) + (2 \times 1) + (1 \times 8) = 8 + 2 + 8 = 18 $$
• Magnitude of Line 1 vector: $$ \sqrt{2^2 + 2^2 + 1^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3 $$
• Magnitude of Line 2 vector: $$ \sqrt{4^2 + 1^2 + 8^2} = \sqrt{16 + 1 + 64} = \sqrt{81} = 9 $$
Substitute these calculated values back into the cosine equation: $$ \cos\theta = \frac{18}{3 \times 9} = \frac{18}{27} $$ Simplifying the fraction by dividing both numbers by 9: $$ \cos\theta = \frac{2}{3} \implies \theta = \cos^{-1}\left(\frac{2}{3}\right) $$
Step 4: Final Answer:
The angle between the two lines is $\cos^{-1}\left(\frac{2}{3}\right)$, which corresponds to option (C).
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