Equation Of A Line In Space

Question

Consider the line passing through the point (2, -1, 3) and parallel to the vector 4i + 0j - 3k in three-dimensional space.

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The relevant standard form is the symmetric (Cartesian) equation of a line through a point (x_0, y_0, z_0) with direction ratios (a, b, c): (x-x_0)/a = (y-y_0)/b = (z-z_0)/c. This representation is a JEE Advanced staple for describing lines in space. Here the point is (2, -1, 3), so x_0 = 2, y_0 = -1, z_0 = 3, and the direction ratios come straight from the parallel vector, giving (a, b, c) = (4, 0, -3). Substituting yields (x-2)/4 = (y+1)/0 = (z-3)/(-3); the zero denominator is conventional shorthand meaning y + 1 = 0 along the line. Option two negates the coordinates of the point incorrectly. Option three uses the point's coordinates as direction ratios, confusing position with direction. Option four scrambles point and direction values entirely. This is the direct application of the point-direction line equation. Plausibility check: substituting the given point makes every fraction equal to zero simultaneously, confirming the point lies on the line as required.

Equation Of A Line In Space

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