Consider a line \(\vec{r}= (\hat{\imath} + 4\hat{\jmath}) +\lambda(2\hat{\imath} - 2\hat{\jmath} + 3\hat{k})\), then which of the following statements are correct?
(A) it passes through point (9, −4, 12)
(B) it passes through point (1, 4, −1)
(C) its direction cosines are \(\displaystyle\frac{2}{\sqrt{17}},\frac{-2}{\sqrt{17}},\frac{3}{\sqrt{17}}\)
(D) its Cartesian equation is \(\displaystyle\frac{x-1}{2}=\frac{y-4}{-2}=\frac{z}{3}\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (C) and (D) only
The line passes through the point (1, 4, 0) and is parallel to \(2\hat{\imath} - 2\hat{\jmath} + 3\hat{k}\), so its direction ratios are 2, −2, 3. A general point on it is
\( \displaystyle (x, y, z) = (1 + 2\lambda,\ 4 - 2\lambda,\ 3\lambda) \)
So the line passes through (1, 4, 0), its direction ratios are 2, −2, 3 and a general point is \((1 + 2\lambda,\ 4 - 2\lambda,\ 3\lambda)\).
Checking the statements
(A) it passes through point (9, −4, 12) — Correct. Putting \(\lambda = 4\):
\(1 + 8 = 9,\quad 4 - 8 = -4,\quad 3(4) = 12\)
(B) it passes through point (1, 4, −1) — Incorrect. x = 1 needs \(\lambda = 0\), which gives the point (1, 4, 0), not (1, 4, −1).
(C) its direction cosines are \(\frac{2}{\sqrt{17}},\frac{-2}{\sqrt{17}},\frac{3}{\sqrt{17}}\) — Correct. Direction cosines are the direction ratios divided by
\(\sqrt{2^2 + (-2)^2 + 3^2} = \sqrt{17}\)
(D) its Cartesian equation is \(\frac{x-1}{2}=\frac{y-4}{-2}=\frac{z}{3}\) — Correct. The Cartesian form of the line through \((x_1, y_1, z_1)\) with direction ratios a, b, c is
\(\displaystyle \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}\)
With (1, 4, 0) and 2, −2, 3:
\(\displaystyle \frac{x - 1}{2} = \frac{y - 4}{-2} = \frac{z}{3}\)