Which of the following statements is/are TRUE?
(A) The angle between the lines whose direction ratios are proportional to (4, -3, 5) and (3, 4, 5) is \( 60^\circ \).
(B) If A(1,2,3) and B(2, 0, 5) are two points on a line, then its direction ratios are proportional to 1, -2, 2.
(C) The direction cosines of z-axis are 0, 0, 1.
(D) The vector equation of a line which passes through the point with position vector \( 3\hat{\imath}-\hat{\jmath}+\hat{k} \) and is in the direction of \( 2\hat{\imath}+3\hat{\jmath}+\hat{k} \) is \( \vec{r}=\left(2\hat{\imath}+3\hat{\jmath}+\hat{k}\right)+\lambda\left(3\hat{\imath}-\hat{\jmath}+\hat{k}\right) \), where \( \lambda \) is a parameter.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (B) and (C) only
The angle between lines with direction ratios (4, −3, 5) and (3, 4, 5):
\( \displaystyle \cos\theta=\frac{|(4)(3)+(-3)(4)+(5)(5)|}{\sqrt{16+9+25}\,\sqrt{9+16+25}} \)
\( \displaystyle =\frac{25}{\sqrt{50}\,\sqrt{50}} \)
\( \displaystyle =\frac{1}{2} \)
\( \theta=60^\circ \)
The direction ratios of the line through A(1, 2, 3) and B(2, 0, 5) are the differences of the coordinates:
\( (2-1,\ 0-2,\ 5-3)=(1,-2,2) \)
The z-axis makes angles \( 90^\circ \), \( 90^\circ \) and \( 0^\circ \) with the x-, y- and z-axes, so its direction cosines are \( \cos 90^\circ,\ \cos 90^\circ,\ \cos 0^\circ \), that is 0, 0, 1.
A line through the point \( \vec{a} \) in the direction \( \vec{b} \) is \( \vec{r}=\vec{a}+\lambda\vec{b} \). With the point \( 3\hat{\imath}-\hat{\jmath}+\hat{k} \) and the direction \( 2\hat{\imath}+3\hat{\jmath}+\hat{k} \):
\( \displaystyle \vec{r}=\left(3\hat{\imath}-\hat{\jmath}+\hat{k}\right)+\lambda\left(2\hat{\imath}+3\hat{\jmath}+\hat{k}\right) \)
So the angle is \( 60^\circ \), the direction ratios are 1, −2, 2, the direction cosines of the z-axis are 0, 0, 1, and the line is \( \vec{r}=\left(3\hat{\imath}-\hat{\jmath}+\hat{k}\right)+\lambda\left(2\hat{\imath}+3\hat{\jmath}+\hat{k}\right) \).
Checking the statements
(A) The angle between the lines whose direction ratios are proportional to (4, -3, 5) and (3, 4, 5) is \( 60^\circ \). — Correct. \( \cos\theta=\frac{1}{2} \), so \( \theta=60^\circ \).
(B) If A(1,2,3) and B(2, 0, 5) are two points on a line, then its direction ratios are proportional to 1, -2, 2. — Correct. The differences are 2 − 1, 0 − 2, 5 − 3.
(C) The direction cosines of z-axis are 0, 0, 1. — Correct. They are \( \cos 90^\circ,\ \cos 90^\circ,\ \cos 0^\circ \).
(D) The vector equation of a line which passes through the point with position vector \( 3\hat{\imath}-\hat{\jmath}+\hat{k} \) and is in the direction of \( 2\hat{\imath}+3\hat{\jmath}+\hat{k} \) is \( \vec{r}=\left(2\hat{\imath}+3\hat{\jmath}+\hat{k}\right)+\lambda\left(3\hat{\imath}-\hat{\jmath}+\hat{k}\right) \), where \( \lambda \) is a parameter. — Incorrect. The point and the direction are interchanged; the point \( 3\hat{\imath}-\hat{\jmath}+\hat{k} \) must come first.