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One vertex of a rectangular parallelopiped is at the origin O and the lengths of its edges along x, y and z axes are 3, 4 and 5 units respectively. Let P be the vertex (3, 4, 5). Then the shortest distance between the diagonal OP and an edge parallel to z axis, not passing through O or P is :?
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One vertex of a rectangular parallelopiped is at the origin O and the ...
Given information:
- One vertex of the rectangular parallelepiped is at the origin O.
- The lengths of its edges along the x, y, and z axes are 3, 4, and 5 units, respectively.
- Let P be the vertex (3, 4, 5).
To find:
- The shortest distance between the diagonal OP and an edge parallel to the z-axis, not passing through O or P.
Solution:
To find the shortest distance between the diagonal OP and an edge parallel to the z-axis, we can use the concept of vector projection.
Step 1: Finding the direction vector of the diagonal OP
- The position vector OP can be obtained by subtracting the position vector of O from the position vector of P.
- The position vector of O is (0, 0, 0), and the position vector of P is (3, 4, 5).
- Therefore, the position vector OP is (3, 4, 5) - (0, 0, 0) = (3, 4, 5).
Step 2: Finding the direction vector of the edge parallel to the z-axis
- Since the edge is parallel to the z-axis, its direction vector will have the form (0, 0, k), where k is a constant.
- The length of the edge along the z-axis is 5 units, so the direction vector will be (0, 0, 5).
Step 3: Finding the projection of the vector OP onto the direction vector of the edge
- The projection of a vector a onto a vector b is given by the formula: projba = (a · b / |b|²) * b, where · denotes dot product and |b| denotes the magnitude of b.
- In this case, the vector a is OP = (3, 4, 5) and the vector b is (0, 0, 5).
- The dot product of OP and (0, 0, 5) is (3 * 0) + (4 * 0) + (5 * 5) = 25.
- The magnitude of (0, 0, 5) is √(0² + 0² + 5²) = √25 = 5.
- Therefore, the projection of OP onto the direction vector of the edge is (25 / 25) * (0, 0, 5) = (0, 0, 5).
Step 4: Calculating the shortest distance
- The shortest distance between the diagonal OP and the edge parallel to the z-axis can be found by subtracting the projection of OP onto the direction vector of the edge from OP.
- (3, 4, 5) - (0, 0, 5) = (3, 4, 5) - (0, 0, 5) = (3, 4, 0).
- The magnitude of (3, 4, 0) is √(3² + 4² + 0²) = √25 = 5.
Answer:
- Therefore, the shortest distance between the diagonal
- One vertex of the rectangular parallelepiped is at the origin O.
- The lengths of its edges along the x, y, and z axes are 3, 4, and 5 units, respectively.
- Let P be the vertex (3, 4, 5).
To find:
- The shortest distance between the diagonal OP and an edge parallel to the z-axis, not passing through O or P.
Solution:
To find the shortest distance between the diagonal OP and an edge parallel to the z-axis, we can use the concept of vector projection.
Step 1: Finding the direction vector of the diagonal OP
- The position vector OP can be obtained by subtracting the position vector of O from the position vector of P.
- The position vector of O is (0, 0, 0), and the position vector of P is (3, 4, 5).
- Therefore, the position vector OP is (3, 4, 5) - (0, 0, 0) = (3, 4, 5).
Step 2: Finding the direction vector of the edge parallel to the z-axis
- Since the edge is parallel to the z-axis, its direction vector will have the form (0, 0, k), where k is a constant.
- The length of the edge along the z-axis is 5 units, so the direction vector will be (0, 0, 5).
Step 3: Finding the projection of the vector OP onto the direction vector of the edge
- The projection of a vector a onto a vector b is given by the formula: projba = (a · b / |b|²) * b, where · denotes dot product and |b| denotes the magnitude of b.
- In this case, the vector a is OP = (3, 4, 5) and the vector b is (0, 0, 5).
- The dot product of OP and (0, 0, 5) is (3 * 0) + (4 * 0) + (5 * 5) = 25.
- The magnitude of (0, 0, 5) is √(0² + 0² + 5²) = √25 = 5.
- Therefore, the projection of OP onto the direction vector of the edge is (25 / 25) * (0, 0, 5) = (0, 0, 5).
Step 4: Calculating the shortest distance
- The shortest distance between the diagonal OP and the edge parallel to the z-axis can be found by subtracting the projection of OP onto the direction vector of the edge from OP.
- (3, 4, 5) - (0, 0, 5) = (3, 4, 5) - (0, 0, 5) = (3, 4, 0).
- The magnitude of (3, 4, 0) is √(3² + 4² + 0²) = √25 = 5.
Answer:
- Therefore, the shortest distance between the diagonal
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One vertex of a rectangular parallelopiped is at the origin O and the lengths of its edges along x, y and z axes are 3, 4 and 5 units respectively. Let P be the vertex (3, 4, 5). Then the shortest distance between the diagonal OP and an edge parallel to z axis, not passing through O or P is :?
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One vertex of a rectangular parallelopiped is at the origin O and the lengths of its edges along x, y and z axes are 3, 4 and 5 units respectively. Let P be the vertex (3, 4, 5). Then the shortest distance between the diagonal OP and an edge parallel to z axis, not passing through O or P is :? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about One vertex of a rectangular parallelopiped is at the origin O and the lengths of its edges along x, y and z axes are 3, 4 and 5 units respectively. Let P be the vertex (3, 4, 5). Then the shortest distance between the diagonal OP and an edge parallel to z axis, not passing through O or P is :? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for One vertex of a rectangular parallelopiped is at the origin O and the lengths of its edges along x, y and z axes are 3, 4 and 5 units respectively. Let P be the vertex (3, 4, 5). Then the shortest distance between the diagonal OP and an edge parallel to z axis, not passing through O or P is :?.
One vertex of a rectangular parallelopiped is at the origin O and the lengths of its edges along x, y and z axes are 3, 4 and 5 units respectively. Let P be the vertex (3, 4, 5). Then the shortest distance between the diagonal OP and an edge parallel to z axis, not passing through O or P is :? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about One vertex of a rectangular parallelopiped is at the origin O and the lengths of its edges along x, y and z axes are 3, 4 and 5 units respectively. Let P be the vertex (3, 4, 5). Then the shortest distance between the diagonal OP and an edge parallel to z axis, not passing through O or P is :? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for One vertex of a rectangular parallelopiped is at the origin O and the lengths of its edges along x, y and z axes are 3, 4 and 5 units respectively. Let P be the vertex (3, 4, 5). Then the shortest distance between the diagonal OP and an edge parallel to z axis, not passing through O or P is :?.
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One vertex of a rectangular parallelopiped is at the origin O and the lengths of its edges along x, y and z axes are 3, 4 and 5 units respectively. Let P be the vertex (3, 4, 5). Then the shortest distance between the diagonal OP and an edge parallel to z axis, not passing through O or P is :?, a detailed solution for One vertex of a rectangular parallelopiped is at the origin O and the lengths of its edges along x, y and z axes are 3, 4 and 5 units respectively. Let P be the vertex (3, 4, 5). Then the shortest distance between the diagonal OP and an edge parallel to z axis, not passing through O or P is :? has been provided alongside types of One vertex of a rectangular parallelopiped is at the origin O and the lengths of its edges along x, y and z axes are 3, 4 and 5 units respectively. Let P be the vertex (3, 4, 5). Then the shortest distance between the diagonal OP and an edge parallel to z axis, not passing through O or P is :? theory, EduRev gives you an
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