Find a vector form equation for the plane that passes through the poin

1.	Find a vector form equation for the plane that passes through the point (2,-1,3) and is parallel to the plane 5x - 2y + z - 2 = 0.
Answer» Since, required plane is parallel to the plane 5x - 2y + z - 2 = 0 \(\therefore\) Normal of given plane is perpendicular to the required plane. \(\therefore\) Direction ratios of normal of required plane are 5, -2, 1 (\(\because\) Direction ratios of given plane are 5, -2, 1) \(\because\) Given plane passes through point (2, -1, 3) and having direction ratios of normal 5, -2, 1 \(\therefore\) Equation of required plane is 5(x - 2) - 2(y + 1) + 1(z - 3) = 0 ⇒ 5x - 2y + z - 10 - 2 - 3 = 0 ⇒ 5x - 27 + z - 15 = 0 Hence, equation of required plane is 5x - 2y + z - 15 = 0

Find a vector form equation for the plane that passes through the point (2,-1,3) and is parallel to the plane 5x - 2y + z - 2 = 0.

Answer»

Since, required plane is parallel to the plane 5x - 2y + z - 2 = 0

\(\therefore\) Normal of given plane is perpendicular to the required plane.

\(\therefore\) Direction ratios of normal of required plane are 5, -2, 1

(\(\because\) Direction ratios of given plane are 5, -2, 1)

\(\because\) Given plane passes through point (2, -1, 3) and having direction ratios of normal 5, -2, 1

\(\therefore\) Equation of required plane is

5(x - 2) - 2(y + 1) + 1(z - 3) = 0

⇒ 5x - 2y + z - 10 - 2 - 3 = 0

⇒ 5x - 27 + z - 15 = 0

Hence, equation of required plane is 5x - 2y + z - 15 = 0

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