Question

Find the vector equation of the line passing through the point $2\hat i + 3\hat j + \hat k$ and parallel to the vector $4\hat i - 2\hat j + 3\hat k$.

Answer 1

In the problem, we are required to find the vector equation of the straight line where a point lying on the line is given to us along with the direction vector of the line. We must know the formula for writing the vector equation of a line given a point lying on it and the direction vector to solve the question. We just substitute the values of the point lying on the line and the vector to which the given line is parallel in order to find the vector equation of the required line.

Complete step by step answer:
So, the point $2\hat i + 3\hat j + \hat k$ lies on the required straight line. Also, the straight line is parallel to the vector $4\hat i - 2\hat j + 3\hat k$. We know that the formula for finding the vector equation of a straight line where a point lying on the line and the direction vector is given to us is $\vec r + \lambda \vec l$ where $\vec r$ is the position vector of the point lying on the line and $\vec l$ is the direction vector of the straight line.

We know that the required line is parallel to the vector $4\hat i - 2\hat j + 3\hat k$. As parallel vectors have the same direction ratios, so the direction ratio of the required line is the same as the direction ratio of the vector $4\hat i - 2\hat j + 3\hat k$. Hence, $4\hat i - 2\hat j + 3\hat k$ is the direction vector of the required line. Now, substituting the values of the point lying on the line and the direction vector, we get,
$2\hat i + 3\hat j + \hat k + \lambda \left( {4\hat i - 2\hat j + 3\hat k} \right)$
Now, separating the rectangular coordinate components, we get,
$\therefore \left( {2 + 4\lambda } \right)\hat i + \left( {3 - 2\lambda } \right)\hat j + \left( {1 + 3\lambda } \right)\hat k$

Therefore, the vector equation of the required line through the point $2\hat i + 3\hat j + \hat k$ and parallel to the vector $4\hat i - 2\hat j + 3\hat k$ is $\left( {2 + 4\lambda } \right)\hat i + \left( {3 - 2\lambda } \right)\hat j + \left( {1 + 3\lambda } \right)\hat k$.

Note: We must have knowledge about the terms direction ratios and directional cosines in order to solve such questions. We should know the method of finding the direction ratios of a vector. The fact that parallel vectors have the same direction ratios must be remembered. The position vector of a point is the vector joining that particular point to the origin.