Find the vector and parametric equations of the line through A(1,2,1) and B(3,-1

Question

Find the vector and parametric equations of the line through A(1,2,1) and B(3,-1,4).

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Explanation / Answer

Answer; The parallel line through the point has the vector equation L = < 0, 1, 2 > + t , and we have to determine a, b, and c. ( 1 ) L is parallel to the plane x + y + z = 2 L is perpendicular to n = < 1, 1, 1 >, the normal to the plane. L and n are perpendicular Thus a + b + c = 0. ( 2 ) L and the line < 1, 1, 0 > + t < 1, -1, 2 > are perpendicular Thus a - b + 2c = 0 We have two equations with three variables: a + b = -c a - b = -2c Solving for a and b in terms of c : 2a = -3c a = -3c/2 2b = c b = c/2 Take c = -2. Then = < 3, -1, -2 > Thus the vector equation of the line is : L = < 0, 1, 2 > + t < 3, -1, -2 > and the parametric equations are: x = 3t, y = 1 - t, z = 2 - 2t

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