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The coordinates of A,B,C are A (–1,2,–3), B(5,0,–6), C(0,4,–1). The direction cosines of the internal beisector of angle BAC are proportional to
- a)6,–2,13
- b)21,2,2
- c)26,–4,6
- d)25,8,5
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
The coordinates of A,B,C are A (–1,2,–3), B(5,0,–6), C(0,4,–1). The di...
Understanding the Problem
To find the direction cosines of the internal bisector of angle BAC, we first need to find the vectors AB and AC. The coordinates of the points are:
- A (–1, 2, –3)
- B (5, 0, –6)
- C (0, 4, –1)
Step 1: Calculate Vectors AB and AC
- Vector AB = B - A = (5 - (–1), 0 - 2, –6 - (–3)) = (6, –2, –3)
- Vector AC = C - A = (0 - (–1), 4 - 2, –1 - (–3)) = (1, 2, 2)
Step 2: Calculate Magnitudes of Vectors
- Magnitude of AB = √(6^2 + (–2)^2 + (–3)^2) = √(36 + 4 + 9) = √49 = 7
- Magnitude of AC = √(1^2 + 2^2 + 2^2) = √(1 + 4 + 4) = √9 = 3
Step 3: Find the Direction Cosines of the Bisector
The direction cosines of the angle bisector can be found using the formula:
Direction Cosines = (magnitude of AC * AB + magnitude of AB * AC) / (magnitude of AB + magnitude of AC)
Calculating the components:
- x-component = (3 * 6 + 7 * 1) / (7 + 3) = (18 + 7) / 10 = 25/10 = 2.5
- y-component = (3 * (–2) + 7 * 2) / (10) = (–6 + 14) / 10 = 8/10 = 0.8
- z-component = (3 * (–3) + 7 * 2) / (10) = (–9 + 14) / 10 = 5/10 = 0.5
Step 4: Scale to Get Proportional Values
To represent the direction cosines proportionally, multiply by 10:
- x = 25
- y = 8
- z = 5
Thus, the direction cosines of the internal bisector of angle BAC are proportional to 25, 8, 5, which corresponds to option 'D'.
To find the direction cosines of the internal bisector of angle BAC, we first need to find the vectors AB and AC. The coordinates of the points are:
- A (–1, 2, –3)
- B (5, 0, –6)
- C (0, 4, –1)
Step 1: Calculate Vectors AB and AC
- Vector AB = B - A = (5 - (–1), 0 - 2, –6 - (–3)) = (6, –2, –3)
- Vector AC = C - A = (0 - (–1), 4 - 2, –1 - (–3)) = (1, 2, 2)
Step 2: Calculate Magnitudes of Vectors
- Magnitude of AB = √(6^2 + (–2)^2 + (–3)^2) = √(36 + 4 + 9) = √49 = 7
- Magnitude of AC = √(1^2 + 2^2 + 2^2) = √(1 + 4 + 4) = √9 = 3
Step 3: Find the Direction Cosines of the Bisector
The direction cosines of the angle bisector can be found using the formula:
Direction Cosines = (magnitude of AC * AB + magnitude of AB * AC) / (magnitude of AB + magnitude of AC)
Calculating the components:
- x-component = (3 * 6 + 7 * 1) / (7 + 3) = (18 + 7) / 10 = 25/10 = 2.5
- y-component = (3 * (–2) + 7 * 2) / (10) = (–6 + 14) / 10 = 8/10 = 0.8
- z-component = (3 * (–3) + 7 * 2) / (10) = (–9 + 14) / 10 = 5/10 = 0.5
Step 4: Scale to Get Proportional Values
To represent the direction cosines proportionally, multiply by 10:
- x = 25
- y = 8
- z = 5
Thus, the direction cosines of the internal bisector of angle BAC are proportional to 25, 8, 5, which corresponds to option 'D'.
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The coordinates of A,B,C are A (–1,2,–3), B(5,0,–6), C(0,4,–1). The direction cosines of the internal beisector of angle BAC are proportional toa)6,–2,13b)21,2,2c)26,–4,6d)25,8,5Correct answer is option 'D'. Can you explain this answer?
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The coordinates of A,B,C are A (–1,2,–3), B(5,0,–6), C(0,4,–1). The direction cosines of the internal beisector of angle BAC are proportional toa)6,–2,13b)21,2,2c)26,–4,6d)25,8,5Correct answer is option 'D'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The coordinates of A,B,C are A (–1,2,–3), B(5,0,–6), C(0,4,–1). The direction cosines of the internal beisector of angle BAC are proportional toa)6,–2,13b)21,2,2c)26,–4,6d)25,8,5Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The coordinates of A,B,C are A (–1,2,–3), B(5,0,–6), C(0,4,–1). The direction cosines of the internal beisector of angle BAC are proportional toa)6,–2,13b)21,2,2c)26,–4,6d)25,8,5Correct answer is option 'D'. Can you explain this answer?.
The coordinates of A,B,C are A (–1,2,–3), B(5,0,–6), C(0,4,–1). The direction cosines of the internal beisector of angle BAC are proportional toa)6,–2,13b)21,2,2c)26,–4,6d)25,8,5Correct answer is option 'D'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The coordinates of A,B,C are A (–1,2,–3), B(5,0,–6), C(0,4,–1). The direction cosines of the internal beisector of angle BAC are proportional toa)6,–2,13b)21,2,2c)26,–4,6d)25,8,5Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The coordinates of A,B,C are A (–1,2,–3), B(5,0,–6), C(0,4,–1). The direction cosines of the internal beisector of angle BAC are proportional toa)6,–2,13b)21,2,2c)26,–4,6d)25,8,5Correct answer is option 'D'. Can you explain this answer?.
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