Which of the following statements are true?
(A) The vector equation of the line through the point (5, 2, −4) and parallel to the vector \(3\hat{\imath} + 2\hat{\jmath} - 8\hat{k}\) is
\(\vec{r} = (5\hat{\imath} + 2\hat{\jmath} - 4\hat{k}) + \lambda(3\hat{\imath} + 2\hat{\jmath} - 8\hat{k})\)
(B) Vector form of the equation of line \(\displaystyle\frac{x-5}{3} = \frac{y+4}{7} = \frac{z-6}{2}\) is \(\vec{r} = (5\hat{\imath} - 4\hat{\jmath} + 6\hat{k}) + \lambda(3\hat{\imath} + 7\hat{\jmath} + 2\hat{k})\)
(C) The direction cosines of z-axis are (1, 1, 0).
(D) If a line has direction ratios 2, −1, −2, then its direction cosines are −2/3, −1/3, −2/3.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) and (B) only
The line through the point with position vector \(\vec{a}\) and parallel to \(\vec{b}\) is \(\vec{r} = \vec{a} + \lambda\vec{b}\). The line \(\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}\) passes through \((x_1, y_1, z_1)\) with direction ratios a, b, c. For the line in (B):
point \((5, -4, 6)\), direction \(3\hat{\imath} + 7\hat{\jmath} + 2\hat{k}\)
The z-axis has angles 90°, 90°, 0° with the axes:
direction cosines \(= (\cos 90^\circ, \cos 90^\circ, \cos 0^\circ) = (0, 0, 1)\)
For direction ratios 2, −1, −2, divide by the magnitude:
\(\sqrt{2^2 + (-1)^2 + (-2)^2} = \sqrt{9} = 3\)
\( \displaystyle \text{direction cosines} = \pm\left(\frac{2}{3}, -\frac{1}{3}, -\frac{2}{3}\right) \)
So the line in (B) passes through (5, −4, 6) along \(3\hat{\imath} + 7\hat{\jmath} + 2\hat{k}\), the z-axis has direction cosines (0, 0, 1), and the line in (D) has direction cosines \(\pm\left(\frac{2}{3}, -\frac{1}{3}, -\frac{2}{3}\right)\).
Checking the statements
(A) The vector equation of the line through the point (5, 2, −4) and parallel to the vector \(3\hat{\imath} + 2\hat{\jmath} - 8\hat{k}\) is \(\vec{r} = (5\hat{\imath} + 2\hat{\jmath} - 4\hat{k}) + \lambda(3\hat{\imath} + 2\hat{\jmath} - 8\hat{k})\) — Correct. It is \(\vec{r} = \vec{a} + \lambda\vec{b}\) with \(\vec{a} = 5\hat{\imath} + 2\hat{\jmath} - 4\hat{k}\).
(B) Vector form of the equation of line \(\frac{x-5}{3} = \frac{y+4}{7} = \frac{z-6}{2}\) is \(\vec{r} = (5\hat{\imath} - 4\hat{\jmath} + 6\hat{k}) + \lambda(3\hat{\imath} + 7\hat{\jmath} + 2\hat{k})\) — Correct. Point (5, −4, 6), direction (3, 7, 2).
(C) The direction cosines of z-axis are (1, 1, 0). — Incorrect. They are (0, 0, 1); also \(1^2 + 1^2 + 0^2 = 2 \ne 1\).
(D) If a line has direction ratios 2, −1, −2, then its direction cosines are −2/3, −1/3, −2/3. — Incorrect. They are \(\pm\left(\frac{2}{3}, -\frac{1}{3}, -\frac{2}{3}\right)\); only the first sign is changed in the statement.