Vectors
Vectors
Allen Star Batch
Grade 12
Question:
A line passes through the points whose position vectors are $\vec{r} + \vec{j} - 2\vec{k}$ and $\vec{r} - 3\vec{j} + \vec{k}$. The position vector of a point on it at a unit distance from the first point is:
$\frac{1}{3}(5\vec{i} + \vec{j} - 7\vec{k})$
$\frac{1}{5}(5\vec{i} + 9\vec{j} - 13\vec{k})$
$\vec{i} - 4\vec{j} + 3\vec{k}$
$\frac{1}{\sqrt{29}}\left(2\vec{i} - 4\vec{j} + 3\vec{k}\right)$
Step-by-Step Solution
Key Concept: Parametric form of a line uses a base point and a direction vector multiplied by a scalar parameter.
The parametric equation of a line through point $(x_0, y_0, z_0)$ with direction vector $(l, m, n)$ is given by $\vec{r} = \vec{r}_0 + \lambda \vec{d}$. Substitute the given point and direction to obtain $\vec{r} = (\vec{i} + \vec{j} - 2\vec{k}) + \lambda\left(\frac{4}{5}\vec{i} - \frac{3}{5}\vec{k}\right)$. Place $\lambda = \pm 1$ as specified in the problem to find specific points on the line.
Correct Answer: 1,2