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# Condition for a line to lie in a plane

We observe that a straight line will lie in a plane if every point on the line, lie in the plane and the normal to the plane is perpendicular to the line.

Condition for a line to lie in a plane

We observe that a straight line will lie in a plane if every point on the line, lie in the plane and the normal to the plane is perpendicular to the line.

i) If the line lies in the plane  = d , then ⋅ = d and . = 0

ii) if the line lies in the plane Ax + By + Cz + D = 0 , then

Ax1 + By1 + Cz1 + D = 0 and aA + bB + cC = 0

### Example 6.45

Verify whether the line lies in the plane 5x y + z = 8 .

### Solution

Here, ( x1, y1, z1 ) = (3, 4, −3) and direction ratios of the given straight line are (a,b, c) = (−4, −7,12) . Direction ratios of the normal to the given plane are ( A, B,C ) = (5, −1,1) .

We observe that, the given point ( x1, y1, z1 ) = (3, 4, −3) satisfies the given plane 5x y + z = 8

Next, aA + bB + cC = (−4)(5) + (−7)(−1) + (12)(1) = −1 ≠ 0 . So, the normal to the plane is not perpendicular to the line. Hence, the given line does not lie in the plane.

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12th Mathematics : UNIT 6 : Applications of Vector Algebra : Condition for a line to lie in a plane |