Question

The measure of the angle between the lines $x = k - 1, y = 2k + 1, z = 2k + 3, k \in \mathbb{R}$ and $\frac{x+1}{2} = \frac{y-2}{1} = \frac{z-1}{2}$ is

• A. $\cos^{-1}\left(\frac{2}{3}\right)$
• B. $\cos^{-1}\left(\frac{8}{9}\right)$
• C. $\cos^{-1}\left(\frac{5}{12}\right)$
• D. $\sin^{-1}\left(\frac{8}{9}\right)$

Hint:

Always convert parametric line equations into symmetric (Cartesian) form to reliably read off the direction ratios from the denominators. Ensure the coefficients of $x, y, z$ in the numerators are all $1$ before reading the denominators.

Answer 1

The Correct Option is B

To find the angle between the given lines, we first need to understand the representations of both lines and how they relate to vectors.

Let's consider the two lines:

• Line 1: Given as \(x = k - 1, y = 2k + 1, z = 2k + 3\), where \(k \in \mathbb{R}\). This is a parametric form of a line through the point \((k-1, 2k+1, 2k+3)\) and can be written in vector form: 
  \(\mathbf{r}_1 = (k-1, 2k+1, 2k+3) = (1, 2, 2) + k(1, 2, 2)\).
• Line 2: Given in symmetric form as \(\frac{x+1}{2} = \frac{y-2}{1} = \frac{z-1}{2}\). The direction ratios for this line are \((2, 1, 2)\).

In vector form, Line 2 is represented as \(\mathbf{r}_2 = (x_0, y_0, z_0) + \lambda (2, 1, 2)\).

The direction vectors for these lines can be identified as:

• Direction vector for Line 1: \(\mathbf{a} = (1, 2, 2)\)
• Direction vector for Line 2: \(\mathbf{b} = (2, 1, 2)\)

The angle \(\theta\) between two lines with direction vectors \(\mathbf{a}\) and \(\mathbf{b}\) is given by the formula:

\(\cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}\)

Calculating the dot product \(\mathbf{a} \cdot \mathbf{b}\):

\(\mathbf{a} \cdot \mathbf{b} = (1 \cdot 2) + (2 \cdot 1) + (2 \cdot 2) = 2 + 2 + 4 = 8\)

Now, calculate the magnitudes of \(\mathbf{a}\) and \(\mathbf{b}\):

\(|\mathbf{a}| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3\)

\(|\mathbf{b}| = \sqrt{2^2 + 1^2 + 2^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3\)

Substituting these values into the formula for \(\cos \theta\):

\(\cos \theta = \frac{8}{3 \times 3} = \frac{8}{9}\)

Therefore, the angle between the lines is:

\(\theta = \cos^{-1}\left(\frac{8}{9}\right)\)

Hence, the correct answer is \(\cos^{-1}\left(\frac{8}{9}\right)\).