Question

The angle between a line with direction ratios 2, 2, 1 and a line joining $(3, 1, 4)$ and $(7, 2, 12)$ is

• A. $\cos^{-1}\left(\frac{1}{3}\right)$
• B. $\cos^{-1}\left(\frac{1}{2}\right)$
• C. $\cos^{-1}\left(\frac{2}{3}\right)$
• D. $\cos^{-1}\left(\frac{\sqrt{3}}{2}\right)$

Hint:

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.

Answer 1

The Correct Option is C

Step 1: Get the direction ratios of both lines.
The first line has direction ratios $(2,2,1)$. The second joins $(3,1,4)$ and $(7,2,12)$, so its ratios are $(7-3,\,2-1,\,12-4)=(4,1,8)$.

Step 2: Use the angle formula.
\[ \cos\theta=\frac{|a_1a_2+b_1b_2+c_1c_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\,\sqrt{a_2^2+b_2^2+c_2^2}} \]

Step 3: Plug in the numbers.
Top: $2\cdot4+2\cdot1+1\cdot8=18$. Magnitudes: $\sqrt{4+4+1}=3$ and $\sqrt{16+1+64}=9$. \[ \cos\theta=\frac{18}{3\cdot9}=\frac{2}{3} \]

Step 4: State the angle.
\[ \boxed{\theta=\cos^{-1}\left(\frac{2}{3}\right)} \]