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Which is a wrong statement? The three planes
a1x + b1y + c1z + d1 = 0
a2x + b2y + c2z + d2 = 0
a3x + b3y + c3z + d3 = 0
have a common line of intersection, if
a1x + b1y + c1z + d1 = 0
a2x + b2y + c2z + d2 = 0
a3x + b3y + c3z + d3 = 0
have a common line of intersection, if
- a)
- b)
- c)
all vanish - d)
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
Which is a wrong statement? The three planesa1x + b1y + c1z + d1 = 0a2...
The three planes are
Denote the determinants as follows:
Note that Δ is th e coefficients of x, y and z . Δ1, Δ2 and Δ3, are obtained from Δ by replacing its first column, 2nd column and 3rd column by d1, d2 and d3, respectively.
Also note that if the three planes are to intersect in a line, then no two of them are parallel. Under this case, the line of intersection of any two planes will lie on the third plane. Now the line of intersection of (i) and (ii) is given by
Since line (iv) lies on plane (iii), therefore is perpendicuar to the normal to the plane (iii)
Also since line (iv) lies on plane (iii), therefore the coordinates of any point on the line will satisfy the equation of plane (iii), since
Thus (A) and (R) are the required conditions for the three planes to have a common line of intersection.
In a similar way, it can he shown that Δ = 0 and Δ1 = 0 or Δ = 0 and Δ2 = 0 are also the required conditions.
∴ (b), (c), (d) are correct.
Remark: To prove that three planes intersect in a line, verify that Δ = 0 and any one: of Δ1,Δ2 and Δ
Denote the determinants as follows:
Note that Δ is th e coefficients of x, y and z . Δ1, Δ2 and Δ3, are obtained from Δ by replacing its first column, 2nd column and 3rd column by d1, d2 and d3, respectively.
Also note that if the three planes are to intersect in a line, then no two of them are parallel. Under this case, the line of intersection of any two planes will lie on the third plane. Now the line of intersection of (i) and (ii) is given by
Since line (iv) lies on plane (iii), therefore is perpendicuar to the normal to the plane (iii)
Also since line (iv) lies on plane (iii), therefore the coordinates of any point on the line will satisfy the equation of plane (iii), since
Thus (A) and (R) are the required conditions for the three planes to have a common line of intersection.
In a similar way, it can he shown that Δ = 0 and Δ1 = 0 or Δ = 0 and Δ2 = 0 are also the required conditions.
∴ (b), (c), (d) are correct.
Remark: To prove that three planes intersect in a line, verify that Δ = 0 and any one: of Δ1,Δ2 and Δ
3
becomes zero.Most Upvoted Answer
Which is a wrong statement? The three planesa1x + b1y + c1z + d1 = 0a2...
The three planes are
Denote the determinants as follows:
Note that Δ is th e coefficients of x, y and z . Δ1, Δ2 and Δ3, are obtained from Δ by replacing its first column, 2nd column and 3rd column by d1, d2 and d3, respectively.
Also note that if the three planes are to intersect in a line, then no two of them are parallel. Under this case, the line of intersection of any two planes will lie on the third plane. Now the line of intersection of (i) and (ii) is given by
Since line (iv) lies on plane (iii), therefore is perpendicuar to the normal to the plane (iii)
Also since line (iv) lies on plane (iii), therefore the coordinates of any point on the line will satisfy the equation of plane (iii), since
Thus (A) and (R) are the required conditions for the three planes to have a common line of intersection.
In a similar way, it can he shown that Δ = 0 and Δ1 = 0 or Δ = 0 and Δ2 = 0 are also the required conditions.
∴ (b), (c), (d) are correct.
Remark: To prove that three planes intersect in a line, verify that Δ = 0 and any one: of Δ1,Δ2 and Δ
Denote the determinants as follows:
Note that Δ is th e coefficients of x, y and z . Δ1, Δ2 and Δ3, are obtained from Δ by replacing its first column, 2nd column and 3rd column by d1, d2 and d3, respectively.
Also note that if the three planes are to intersect in a line, then no two of them are parallel. Under this case, the line of intersection of any two planes will lie on the third plane. Now the line of intersection of (i) and (ii) is given by
Since line (iv) lies on plane (iii), therefore is perpendicuar to the normal to the plane (iii)
Also since line (iv) lies on plane (iii), therefore the coordinates of any point on the line will satisfy the equation of plane (iii), since
Thus (A) and (R) are the required conditions for the three planes to have a common line of intersection.
In a similar way, it can he shown that Δ = 0 and Δ1 = 0 or Δ = 0 and Δ2 = 0 are also the required conditions.
∴ (b), (c), (d) are correct.
Remark: To prove that three planes intersect in a line, verify that Δ = 0 and any one: of Δ1,Δ2 and Δ
3
becomes zero.Free Test
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Community Answer
Which is a wrong statement? The three planesa1x + b1y + c1z + d1 = 0a2...
The three planes are
Denote the determinants as follows:
Note that Δ is th e coefficients of x, y and z . Δ1, Δ2 and Δ3, are obtained from Δ by replacing its first column, 2nd column and 3rd column by d1, d2 and d3, respectively.
Also note that if the three planes are to intersect in a line, then no two of them are parallel. Under this case, the line of intersection of any two planes will lie on the third plane. Now the line of intersection of (i) and (ii) is given by
Since line (iv) lies on plane (iii), therefore is perpendicuar to the normal to the plane (iii)
Also since line (iv) lies on plane (iii), therefore the coordinates of any point on the line will satisfy the equation of plane (iii), since
Thus (A) and (R) are the required conditions for the three planes to have a common line of intersection.
In a similar way, it can he shown that Δ = 0 and Δ1 = 0 or Δ = 0 and Δ2 = 0 are also the required conditions.
∴ (b), (c), (d) are correct.
Remark: To prove that three planes intersect in a line, verify that Δ = 0 and any one: of Δ1,Δ2 and Δ
Denote the determinants as follows:
Note that Δ is th e coefficients of x, y and z . Δ1, Δ2 and Δ3, are obtained from Δ by replacing its first column, 2nd column and 3rd column by d1, d2 and d3, respectively.
Also note that if the three planes are to intersect in a line, then no two of them are parallel. Under this case, the line of intersection of any two planes will lie on the third plane. Now the line of intersection of (i) and (ii) is given by
Since line (iv) lies on plane (iii), therefore is perpendicuar to the normal to the plane (iii)
Also since line (iv) lies on plane (iii), therefore the coordinates of any point on the line will satisfy the equation of plane (iii), since
Thus (A) and (R) are the required conditions for the three planes to have a common line of intersection.
In a similar way, it can he shown that Δ = 0 and Δ1 = 0 or Δ = 0 and Δ2 = 0 are also the required conditions.
∴ (b), (c), (d) are correct.
Remark: To prove that three planes intersect in a line, verify that Δ = 0 and any one: of Δ1,Δ2 and Δ
3
becomes zero.
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Which is a wrong statement? The three planesa1x + b1y + c1z + d1 = 0a2x + b2y + c2z + d2 = 0a3x + b3y + c3z + d3 = 0have a common line of intersection, ifa)b)c)all vanishd)Correct answer is option 'A'. Can you explain this answer? for Mathematics 2025 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Which is a wrong statement? The three planesa1x + b1y + c1z + d1 = 0a2x + b2y + c2z + d2 = 0a3x + b3y + c3z + d3 = 0have a common line of intersection, ifa)b)c)all vanishd)Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Mathematics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which is a wrong statement? The three planesa1x + b1y + c1z + d1 = 0a2x + b2y + c2z + d2 = 0a3x + b3y + c3z + d3 = 0have a common line of intersection, ifa)b)c)all vanishd)Correct answer is option 'A'. Can you explain this answer?.
Which is a wrong statement? The three planesa1x + b1y + c1z + d1 = 0a2x + b2y + c2z + d2 = 0a3x + b3y + c3z + d3 = 0have a common line of intersection, ifa)b)c)all vanishd)Correct answer is option 'A'. Can you explain this answer? for Mathematics 2025 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Which is a wrong statement? The three planesa1x + b1y + c1z + d1 = 0a2x + b2y + c2z + d2 = 0a3x + b3y + c3z + d3 = 0have a common line of intersection, ifa)b)c)all vanishd)Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Mathematics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which is a wrong statement? The three planesa1x + b1y + c1z + d1 = 0a2x + b2y + c2z + d2 = 0a3x + b3y + c3z + d3 = 0have a common line of intersection, ifa)b)c)all vanishd)Correct answer is option 'A'. Can you explain this answer?.
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Which is a wrong statement? The three planesa1x + b1y + c1z + d1 = 0a2x + b2y + c2z + d2 = 0a3x + b3y + c3z + d3 = 0have a common line of intersection, ifa)b)c)all vanishd)Correct answer is option 'A'. Can you explain this answer?, a detailed solution for Which is a wrong statement? The three planesa1x + b1y + c1z + d1 = 0a2x + b2y + c2z + d2 = 0a3x + b3y + c3z + d3 = 0have a common line of intersection, ifa)b)c)all vanishd)Correct answer is option 'A'. Can you explain this answer? has been provided alongside types of Which is a wrong statement? The three planesa1x + b1y + c1z + d1 = 0a2x + b2y + c2z + d2 = 0a3x + b3y + c3z + d3 = 0have a common line of intersection, ifa)b)c)all vanishd)Correct answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
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