Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes and .

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Solution

Let the required line be parallel to vector given by,

The position vector of the point (1, 2, 3) is

The equation of line passing through (1, 2, 3) and parallel to is given by,

The equations of the given planes are

The line in equation (1) and plane in equation (2) are parallel. Therefore, the normal to the plane of equation (2) and the given line are perpendicular.

From equations (4) and (5), we obtain

Therefore, the direction ratios of are −3, 5, and 4.

Substituting the value of in equation (1), we obtain

This is the equation of the required line.

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P (2, - 1, 3)

\to r =^i +^j -^k + λ (2^i - 2^j +^k)

\to r = (2^i -^j - 3^k) + μ (^i + 2^j + 2^k)

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\to r = (^i +^j -^k) + λ (2^i - 2^j +^k)

\to r = (2^i -^j - 3^k) + μ (^i + 2^j + 2^k)

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