Question 1:In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin. (a) z = 2 (b) x + y + z = 1 (c) 2x + 3y – z = 5 (d) 5y + 8 = 0
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Question 2:Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector i j k ˆ 3 ˆ 5 ˆ 6 .
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Question 3:Find the Cartesian equation of the following planes: (a) ˆ ˆ ˆ ri j k ( ) 2 (b) ˆ ˆ ˆ ri j k (2 3 4 ) 1 (c) ˆ ˆ ˆ r s ti t j s tk [( 2 ) (3 ) (2 ) ] 15
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Question 4:In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin. (a) 2x + 3y + 4z – 12 = 0 (b) 3y + 4z – 6 = 0 (c) x + y + z = 1 (d) 5y + 8 = 0
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Question 5:Find the vector and cartesian equations of the planes (a) that passes through the point (1, 0, – 2) and the normal to the plane is ˆ ˆ ˆ i jk . (b) that passes through the point (1,4, 6) and the normal vector to the plane is ˆ ˆ ˆ i jk 2 .
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Question 6:Find the equations of the planes that passes through three points. (a) (1, 1, – 1), (6, 4, – 5), (– 4, – 2, 3) (b) (1, 1, 0), (1, 2, 1), (– 2, 2, – 1)
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Question 7:Find the intercepts cut off by the plane 2x + y – z = 5.
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Question 8:Find the equation of the plane with intercept 3 on the y-axis and parallel to ZOX plane.
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Question 9:Find the equation of the plane through the intersection of the planes 3x – y + 2z – 4 = 0 and x + y + z – 2 = 0 and the point (2, 2, 1).
1. In each of the following cases, determine the d
Question 10:Find the vector equation of the plane passing through the intersection of the planes ˆ ˆ ˆ ri jk .(2 2 3 ) 7 , ˆ ˆ ˆ ri jk .(2 5 3 ) 9 and through the point (2, 1, 3).
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Question 11:Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0
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Question 12:Find the angle between the planes whose vector equations are ˆ ˆ ˆ ri j k (2 2 3 ) 5 and ˆ ˆ ˆ ri j k (3 3 5 ) 3 .
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Question 13:In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them. (a) 7x + 5y + 6z + 30 = 0 and 3x – y – 10z + 4 = 0 (b) 2x + y + 3z – 2 = 0 and x – 2y + 5 = 0 (c) 2x – 2y + 4z + 5 = 0 and 3x – 3y + 6z – 1 = 0 (d) 2x – y + 3z – 1 = 0 and 2x – y + 3z + 3 = 0 (e) 4x + 8y + z – 8 = 0 and y + z – 4 = 0
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Question 14:In the following cases, find the distance of each of the given points from the corresponding given plane. Point Plane (a) (0, 0, 0) 3x – 4y + 12 z = 3 (b) (3, – 2, 1) 2x – y + 2z + 3 = 0 (c) (2, 3, – 5) x + 2y – 2z = 9 (d) (– 6, 0, 0) 2x – 3y + 6z – 2 =0
