Question

Find the Cartesian equation of a line passing through (1, -1, 2) and parallel to the line whose equations are x-3/1= y-1/2=z+1/-2 . Also, reduce the equation obtained in vector form.____

Answer 1

To determine the Cartesian equation of a line parallel to X, we use the given direction ratios and the point through which the new line passes, resulting in the equation (x - 1)/1 = (y + 1)/2 = (z - 2)/-2 in symmetric form and vector form r = (1, -1, 2) + t(1, 2, -2).

Step-by-step explanation:

To find the Cartesian equation of a line passing through (1, -1, 2) and parallel to the given line, we must perform the following steps:

1. Identify the direction ratios of the given line, which are the coefficients of the denominators in the given equations: (1, 2, -2).
2. Since the required line is parallel to the given line, it will have the same direction ratios. Therefore, the required line will also have the direction vectors (1, 2, -2).
3. Now, use the point (1, -1, 2) and the direction ratios to write the equation of the required line in the symmetric form:
   (x - 1)/1 = (y + 1)/2 = (z - 2)/-2
4. Finally, rearrange this into the vector form of a line equation:
   r = (1, -1, 2) + t(1, 2, -2), where t is a parameter.

The vector form reduces to:

• x = 1 + t
• y = -1 + 2t
• z = 2 - 2t