Question

The angle between the straight lines \(\frac{x+4}{2}=\frac{y+5}{5}=\frac{z+6}{3} \space and \space \frac{x-4}{10}=\frac{y-5}{2}=\frac{z-6}{-10}\) is.

• A. 45°
• B. 90°
• C. 30°
• D. 60°

Answer 1

The Correct Option is B

To determine the angle between the given lines, we first need the direction ratios of each line. The direction ratios of a line in the format \( \frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c} \) are \( a, b, c \).

For the first line \(\frac{x+4}{2}=\frac{y+5}{5}=\frac{z+6}{3}\), the direction ratios are \(2, 5, 3\).

For the second line \(\frac{x-4}{10}=\frac{y-5}{2}=\frac{z-6}{-10}\), the direction ratios are \(10, 2, -10\).

The angle \( \theta \) between two lines with direction ratios \(a_1, b_1, c_1\) and \(a_2, b_2, c_2\) can be found using the formula:

\( \cos(\theta) = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2}\sqrt{a_2^2 + b_2^2 + c_2^2}} \)

Substituting the direction ratios:

\( \cos(\theta) = \frac{2 \times 10 + 5 \times 2 + 3 \times (-10)}{\sqrt{2^2 + 5^2 + 3^2}\sqrt{10^2 + 2^2 + (-10)^2}} \)

Calculating the dot product:

\( = \frac{20 + 10 - 30}{\sqrt{4 + 25 + 9}\sqrt{100 + 4 + 100}} \)

\( = \frac{0}{\sqrt{38} \times \sqrt{204}} \)

Since the numerator is 0, \( \cos(\theta) = 0 \), implying that \( \theta = 90^\circ \) (right angle).

Thus, the angle between the lines is 90°.