The Cartesian equation of the plane r → · ( j ^ - 2 k ^ ) = 7 is

Given that, the equation of a plane: r → · ( j ^ - 2 k ^ ) = 7 Suppose P x , y , z be any point on the plane. Then its position vector is r → = x i ^ + y j ^ + z k ^ Now, x i ^ + y j ^ + z k ^ · ( j ^ - 2 k ^ ) = 7 ⇒ y - 2 z = 7 [Using i ^ · i ^ = j ^ · j ^ = k ^ · k ^ = 1 ] ⇒ y - 2 z - 7 = 0 , which is the required equation of the plane in Cartesian form.

The x - intercept of the plane is r → · ( 5 j ^ + 11 k ^ ) = 9 .

Given that, the equation of a plane: r → · ( 5 j ^ + 11 k ^ ) = 9 Suppose P x , y , z be any point on the plane. Then its position vector is r → = x i ^ + y j ^ + z k ^ Now, x i ^ + y j ^ + z k ^ · ( 5 j ^ + 11 k ^ ) = 9 ⇒ 5 y + 11 z = 9 [Using i ^ · i ^ = j ^ · j ^ = k ^ · k ^ = 1 ] ⇒ 5 y + 11 z 9 = 9 9 [Dividing with 9 on both sides] ⇒ 5 y 9 + 11 z 9 = 1 ⇒ y 9 5 + z 9 11 = 1 On comparing with intercept form of the plane x a + y b + z c = 1 , we can see that, there is no x - intercept exist i.e., no x component. Hence, the x - intercept of equation of the plane is undefined.

The distance of P ( 2 , 1 , - 1 ) from the plane r → · ( i ^ - 2 j ^ + 4 k ^ ) = 9 is _____ unit(s)

Given equation of plane: r → · ( i ^ - 2 j ^ + 4 k ^ ) = 9 and point P ( 2 , 1 , - 1 ) We know that, The perpendicular distance of a point with position vector a → from the plane r → · n → = d is D = | a → · n → - d | | n → | Here, a → = 2 i ^ + j ^ - k ^ , n → = ( i ^ - 2 j ^ + 4 k ^ ) and d = 9 Therefore, D = 2 i ^ + j ^ - k ^ · ( i ^ - 2 j ^ + 4 k ^ ) - 9 ( i ^ - 2 j ^ + 4 k ^ ) ⇒ D = 2 - 2 - 4 - 9 1 + 4 + 16 ⇒ D = 13 21 units Hence, the distance of P ( 2 , 1 , - 1 ) from the plane r → · ( i ^ - 2 j ^ + 4 k ^ ) = 9 is 13 21 units.