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Find the direction cosines of a line which makes equal angles with all three the coordinate axes.
- a)± 1/√2, ± 1/√2, ± 1/√2
- b)1/√3, 1/√3, 1/√3
- c)1/√2, 1/√2, 1/√2
- d)± 1/√3, ± 1/√3, ± 1/√3
Correct answer is option 'D'. Can you explain this answer?
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Understanding Direction Cosines
Direction cosines are the cosines of the angles that a line makes with the coordinate axes. For a line to make equal angles with all three axes, the direction cosines must be equal in magnitude.
Equal Angles with Axes
If a line makes equal angles with the x, y, and z axes, then we can denote the direction cosines as:
- l = cos(α)
- m = cos(β)
- n = cos(γ)
Where α, β, and γ are the angles with the x, y, and z axes, respectively.
Since the angles are equal, we have:
- l = m = n
Magnitude Condition
The sum of the squares of the direction cosines must equal 1:
- l² + m² + n² = 1
Substituting l = m = n gives:
- 3l² = 1
- l² = 1/3
- l = ±1/√3
Thus, the direction cosines can be expressed as:
- l = ±1/√3
- m = ±1/√3
- n = ±1/√3
Conclusion
The direction cosines that make equal angles with the coordinate axes are:
- ±1/√3, ±1/√3, ±1/√3
This corresponds to option 'D':
- Direction cosines: ±1/√3, ±1/√3, ±1/√3
Therefore, the correct answer is indeed option 'D'.
Direction cosines are the cosines of the angles that a line makes with the coordinate axes. For a line to make equal angles with all three axes, the direction cosines must be equal in magnitude.
Equal Angles with Axes
If a line makes equal angles with the x, y, and z axes, then we can denote the direction cosines as:
- l = cos(α)
- m = cos(β)
- n = cos(γ)
Where α, β, and γ are the angles with the x, y, and z axes, respectively.
Since the angles are equal, we have:
- l = m = n
Magnitude Condition
The sum of the squares of the direction cosines must equal 1:
- l² + m² + n² = 1
Substituting l = m = n gives:
- 3l² = 1
- l² = 1/3
- l = ±1/√3
Thus, the direction cosines can be expressed as:
- l = ±1/√3
- m = ±1/√3
- n = ±1/√3
Conclusion
The direction cosines that make equal angles with the coordinate axes are:
- ±1/√3, ±1/√3, ±1/√3
This corresponds to option 'D':
- Direction cosines: ±1/√3, ±1/√3, ±1/√3
Therefore, the correct answer is indeed option 'D'.
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