Question

The equation of a line passing through the origin and parallel to the line \[ \vec{r} = 3\hat{i} + 4\hat{j} - 5\hat{k} + t(2\hat{i} - \hat{j} + 7\hat{k}), \] where $t$ is a parameter, is:
  (A) $\frac{x}{2} = \frac{y}{-1} = \frac{z}{7}$  (B) $\vec{r} = m(12\hat{i} - 6\hat{j} + 42\hat{k});$ where $m$ is the parameter  (C) $\vec{r} = (12\hat{i} - 6\hat{j} + 42\hat{k}) + s(0\hat{i} - 0\hat{j} + 0\hat{k});$ where $s$ is the parameter  (D) $\frac{x - 3}{3} = \frac{y - 4}{-4} = \frac{z + 5}{0}$  (E) $\frac{x}{3} = \frac{y}{4} = \frac{z}{5}$
Choose the correct answer from the options given below:

• A. (A) and (B) only
• B. (A), (B) and (C) only
• C. (C), (D) and (E) only
• D. (A) only

Answer 1

The Correct Option is A

The given line is represented by the equation:
\[ \vec{r} = 3\hat{i} + 4\hat{j} - 5\hat{k} + t(2\hat{i} - \hat{j} + 7\hat{k}) \]
This line passes through the point \((3, 4, -5)\) and has the direction vector \( \vec{d_1} = 2\hat{i} - \hat{j} + 7\hat{k} \). A line parallel to this line will have the same direction vector.
The requirement is to find the line passing through the origin \((0, 0, 0)\) and parallel to \(\vec{d_1}\).
Hence, the equation of a line through the origin with direction vector \(\vec{d_1}\) is:
\(\vec{r} = t(2\hat{i} - \hat{j} + 7\hat{k})\)
Expressed parametrically, this is:
\[ \frac{x}{2} = \frac{y}{-1} = \frac{z}{7} \](This matches choice (A)).
Additionally, the same line can be represented in terms of any scalar multiple 'm' as:
\[ \vec{r} = m(2\hat{i} - \hat{j} + 7\hat{k}) \]
Let's simplify:
\(m = 6t\) therefore:
\(\vec{r} = 6t(2\hat{i} - \hat{j} + 7\hat{k}) = m(12\hat{i} - 6\hat{j} + 42\hat{k})\).
This matches the given line equation in choice (B).
Therefore correct options are choices (A) and (B).

Answer 2

To find the equation of a line passing through the origin and parallel to the given line, we use the direction vector of the line, which is \(2\hat{i} - \hat{j} + 7\hat{k}\).

(A) The symmetric form \(\frac{x}{2} = \frac{y}{-1} = \frac{z}{7}\) represents a line parallel to the given direction vector and passing through the origin.

(B) The vector equation \(\vec{r} = m(12\hat{i} - 6\hat{j} + 42\hat{k})\) is obtained by multiplying the direction vector by a scalar and also passes through the origin.

(C) The form \(\vec{r} = (12\hat{i} - 6\hat{j} + 42\hat{k}) + s(0\hat{i} - 0\hat{j} + 0\hat{k})\) is incorrect as it introduces an additional term that is not needed for parallelism through the origin.

(D) The given equation does not pass through the origin due to the constants.

(E) This form does not represent a line parallel to the given vector.