Which of the following statements are correct?(A) The angle between t…

The correct answer is Option (1) → (A), (B) and (D) only

For (A), the cosine of the angle between the two directions is

    \( \displaystyle \cos\theta=\frac{(-3)(4)+(5)(5)+(4)(3)}{\sqrt{9+25+16}\,\sqrt{16+25+9}} \)

    \( \displaystyle =\frac{25}{\sqrt{50}\sqrt{50}} \)

    \( \displaystyle =\frac{1}{2} \)

For (C), write the equation in standard form, with coefficient 1 for x, y and z:

    \( \displaystyle 3(x-2)=2\left(y+\frac{1}{2}\right)=2\left(z-\frac{1}{2}\right) \)

    \( \displaystyle \frac{x-2}{\frac{1}{3}}=\frac{y+\frac{1}{2}}{\frac{1}{2}}=\frac{z-\frac{1}{2}}{\frac{1}{2}} \)

The direction ratios are \( \frac{1}{3},\frac{1}{2},\frac{1}{2} \); multiplying by 6, they are proportional to 2, 3, 3.

So the angle in (A) is \( \frac{\pi}{3} \) and the direction ratios in (C) are proportional to 2, 3, 3.

Checking the statements

(A) The angle between the vectors with direction ratios proportional to -3, 5, 4 and 4, 5, 3 is \( \frac{\pi}{3} \). — Correct. \( \cos\theta=\frac{25}{50}=\frac{1}{2} \), so \( \theta=\frac{\pi}{3} \).

(B) The direction cosines of Y-axis is 0, 1, 0. — Correct. The Y-axis makes angles 90°, 0°, 90° with the axes, and cos 90° = 0, cos 0° = 1.

(C) The cartesian equation of a line is \( 3x-6=2y+1=2z-1 \), then its direction ratios are proportional to 2, 2, 3 — Incorrect. They are proportional to 2, 3, 3, not 2, 2, 3.

(D) The vector equation of the line passing through the point \( (2,-1,-1) \) and direction ratios 1, 2, 3 is \( \vec{r}=(2\hat{\imath}-\hat{\jmath}-\hat{k})+\lambda(\hat{\imath}+2\hat{\jmath}+3\hat{k}) \) — Correct. This is \( \vec{r}=\vec{a}+\lambda\vec{b} \) with \( \vec{a} \) the position vector of the point and \( \vec{b} \) along the direction ratios.

Tip: The direction ratios of \( ax+p=by+q=cz+r \) are \( \frac{1}{a},\frac{1}{b},\frac{1}{c} \), not a, b, c: the coefficients go into the denominators.