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If l, m and n are the direction cosines of a line, Direction ratios of the line are the numbers which are
- a)Proportional to the direction cosine l of the line
- b)Inversely Proportional to the direction cosines of the line
- c)Inversely Proportional to the direction cosine l of the line
- d)Proportional to the direction cosines of the line.
Correct answer is option 'D'. Can you explain this answer?
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If l, m and n are the direction cosines of a line, Direction ratios of...
If l, m and n are the direction cosines of a line, Direction ratios of the line are the numbers which are Proportional to the direction cosines of the line.
Most Upvoted Answer
If l, m and n are the direction cosines of a line, Direction ratios of...
Explanation:
To understand why the correct answer is option D, let's first clarify the definitions of direction cosines and direction ratios.
Direction Cosines:
Direction cosines are the cosines of the angles that a line makes with the positive x, y, and z-axes. Let the direction cosines of a line be l, m, and n. These direction cosines satisfy the following conditions:
1. l² + m² + n² = 1 (as the cosines of angles always lie between -1 and 1)
2. l, m, and n can be positive, negative, or zero depending on the direction of the line with respect to the axes.
Direction Ratios:
Direction ratios are the ratios of the distances between the coordinates of any two points on the line along the x, y, and z-axes. Let the direction ratios of a line be a, b, and c. These direction ratios are related to the direction cosines as follows:
1. a = l / √(l² + m² + n²)
2. b = m / √(l² + m² + n²)
3. c = n / √(l² + m² + n²)
The direction ratios are always proportional to the direction cosines.
Explanation of the Correct Answer:
According to the given question, l, m, and n are the direction cosines of the line. We need to determine the relationship between the direction ratios and the direction cosines.
From the above equations relating direction ratios and direction cosines, we can observe that:
- The direction ratios a, b, and c are directly proportional to l, m, and n respectively.
- The proportionality constant is 1 divided by the square root of the sum of squares of the direction cosines.
Hence, the direction ratios are directly proportional to the direction cosines of the line.
Conclusion:
Based on the above explanations, we can conclude that the direction ratios of a line are proportional to the direction cosines of the line. Therefore, the correct answer is option D.
To understand why the correct answer is option D, let's first clarify the definitions of direction cosines and direction ratios.
Direction Cosines:
Direction cosines are the cosines of the angles that a line makes with the positive x, y, and z-axes. Let the direction cosines of a line be l, m, and n. These direction cosines satisfy the following conditions:
1. l² + m² + n² = 1 (as the cosines of angles always lie between -1 and 1)
2. l, m, and n can be positive, negative, or zero depending on the direction of the line with respect to the axes.
Direction Ratios:
Direction ratios are the ratios of the distances between the coordinates of any two points on the line along the x, y, and z-axes. Let the direction ratios of a line be a, b, and c. These direction ratios are related to the direction cosines as follows:
1. a = l / √(l² + m² + n²)
2. b = m / √(l² + m² + n²)
3. c = n / √(l² + m² + n²)
The direction ratios are always proportional to the direction cosines.
Explanation of the Correct Answer:
According to the given question, l, m, and n are the direction cosines of the line. We need to determine the relationship between the direction ratios and the direction cosines.
From the above equations relating direction ratios and direction cosines, we can observe that:
- The direction ratios a, b, and c are directly proportional to l, m, and n respectively.
- The proportionality constant is 1 divided by the square root of the sum of squares of the direction cosines.
Hence, the direction ratios are directly proportional to the direction cosines of the line.
Conclusion:
Based on the above explanations, we can conclude that the direction ratios of a line are proportional to the direction cosines of the line. Therefore, the correct answer is option D.
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If l, m and n are the direction cosines of a line, Direction ratios of the line are the numbers which area)Proportional to the direction cosine l of the lineb)Inversely Proportional to the direction cosines of the linec)Inversely Proportional to the direction cosine l of the lined)Proportional to the direction cosines of the line.Correct answer is option 'D'. Can you explain this answer?
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