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If the vectors 6i - 2j + 3k, 2i + 3j- 6k and 3i + 6j- 2k form a triangle, then it is
- a)Right angled
- b)Obtuse angled
- c)Equilateral
- d)Isosceles
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
If the vectors 6i - 2j + 3k, 2i + 3j- 6k and 3i + 6j- 2k form a triang...
Understanding the Vectors
The given vectors are:
- A = 6i - 2j + 3k
- B = 2i + 3j - 6k
- C = 3i + 6j - 2k
These vectors represent the vertices of a triangle when placed tail-to-head in a coordinate system.
Finding the Lengths of the Sides
To determine the type of triangle formed, we need to calculate the lengths of the sides using the distance formula for vectors.
- Length of side AB:
|AB| = |A - B| = |(6-2)i + (-2-3)j + (3+6)k| = |4i - 5j + 9k|
- Length of side BC:
|BC| = |B - C| = |(2-3)i + (3-6)j + (-6+2)k| = |-1i - 3j - 4k|
- Length of side CA:
|CA| = |C - A| = |(3-6)i + (6+2)j + (-2-3)k| = |-3i + 8j - 5k|
Calculating the Squared Lengths
Instead of calculating the lengths directly, we can compare the squares of the lengths to avoid unnecessary calculations:
- |AB|^2 = (4^2) + (-5^2) + (9^2) = 16 + 25 + 81 = 122
- |BC|^2 = (-1^2) + (-3^2) + (-4^2) = 1 + 9 + 16 = 26
- |CA|^2 = (-3^2) + (8^2) + (-5^2) = 9 + 64 + 25 = 98
Determining the Type of Triangle
To classify the triangle:
- Calculate the sum of the squares of the two shorter sides and compare it to the square of the longest side:
26 + 98 = 124 (sum of squares of shorter sides)
122 (square of the longest side)
Since 124 > 122, the triangle is obtuse.
Conclusion
The triangle formed by the given vectors is obtuse-angled, confirming option 'B' as the correct answer.
The given vectors are:
- A = 6i - 2j + 3k
- B = 2i + 3j - 6k
- C = 3i + 6j - 2k
These vectors represent the vertices of a triangle when placed tail-to-head in a coordinate system.
Finding the Lengths of the Sides
To determine the type of triangle formed, we need to calculate the lengths of the sides using the distance formula for vectors.
- Length of side AB:
|AB| = |A - B| = |(6-2)i + (-2-3)j + (3+6)k| = |4i - 5j + 9k|
- Length of side BC:
|BC| = |B - C| = |(2-3)i + (3-6)j + (-6+2)k| = |-1i - 3j - 4k|
- Length of side CA:
|CA| = |C - A| = |(3-6)i + (6+2)j + (-2-3)k| = |-3i + 8j - 5k|
Calculating the Squared Lengths
Instead of calculating the lengths directly, we can compare the squares of the lengths to avoid unnecessary calculations:
- |AB|^2 = (4^2) + (-5^2) + (9^2) = 16 + 25 + 81 = 122
- |BC|^2 = (-1^2) + (-3^2) + (-4^2) = 1 + 9 + 16 = 26
- |CA|^2 = (-3^2) + (8^2) + (-5^2) = 9 + 64 + 25 = 98
Determining the Type of Triangle
To classify the triangle:
- Calculate the sum of the squares of the two shorter sides and compare it to the square of the longest side:
26 + 98 = 124 (sum of squares of shorter sides)
122 (square of the longest side)
Since 124 > 122, the triangle is obtuse.
Conclusion
The triangle formed by the given vectors is obtuse-angled, confirming option 'B' as the correct answer.
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If the vectors 6i - 2j + 3k, 2i + 3j- 6k and 3i + 6j- 2k form a triangle, then it isa)Right angledb)Obtuse angledc)Equilaterald)IsoscelesCorrect answer is option 'B'. Can you explain this answer? for NEET 2025 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about If the vectors 6i - 2j + 3k, 2i + 3j- 6k and 3i + 6j- 2k form a triangle, then it isa)Right angledb)Obtuse angledc)Equilaterald)IsoscelesCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for NEET 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the vectors 6i - 2j + 3k, 2i + 3j- 6k and 3i + 6j- 2k form a triangle, then it isa)Right angledb)Obtuse angledc)Equilaterald)IsoscelesCorrect answer is option 'B'. Can you explain this answer?.
If the vectors 6i - 2j + 3k, 2i + 3j- 6k and 3i + 6j- 2k form a triangle, then it isa)Right angledb)Obtuse angledc)Equilaterald)IsoscelesCorrect answer is option 'B'. Can you explain this answer? for NEET 2025 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about If the vectors 6i - 2j + 3k, 2i + 3j- 6k and 3i + 6j- 2k form a triangle, then it isa)Right angledb)Obtuse angledc)Equilaterald)IsoscelesCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for NEET 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the vectors 6i - 2j + 3k, 2i + 3j- 6k and 3i + 6j- 2k form a triangle, then it isa)Right angledb)Obtuse angledc)Equilaterald)IsoscelesCorrect answer is option 'B'. Can you explain this answer?.
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