Question 2 of 25intermediate🔧 ApplyShort Answer4 marks

Find the direction cosines of a line which makes equal angles with the coordinate axes.

Correct Answer

Let the line make angles α, β, γ with the x, y, z-axes respectively. Since the line makes equal angles with the coordinate axes, we have α = β = γ.

Let l, m, n be the direction cosines of the line. Then l = cos α, m = cos β, n = cos γ. Since α = β = γ, we have l = m = n.

We know that for direction cosines: l² + m² + n² = 1

Substituting l = m = n: l² + l² + l² = 1 3l² = 1 l² = 1/3 l = ±1/3\sqrt{3}

Therefore, the direction cosines are: (1/3\sqrt{3}, 1/3\sqrt{3}, 1/3\sqrt{3}) or (-1/3\sqrt{3}, -1/3\sqrt{3}, -1/3\sqrt{3})

Exercise: EXERCISE 11.1 | Q: 2 | (Page Page 5)
For More Understanding

Explanation

This question tests understanding of direction cosines and the fundamental relation l² + m² + n² = 1. The key insight is that equal angles with all three axes means all three direction cosines are equal. The textbook context confirms that direction cosines are cosines of angles made with coordinate axes, and that the sum of squares equals 1. The two possible answers arise because a line can make an acute angle or an obtuse angle with each axis - reversing the direction of the line reverses the signs of all direction cosines.

Solution Steps

  1. Step 1: Let α, β, γ be the angles made with x, y, z-axes. Since angles are equal, α = β = γ

  2. Step 2: Direction cosines l = cos α, m = cos β, n = cos γ. Therefore l = m = n

  3. Step 3: Apply the relation l² + m² + n² = 1

  4. Step 4: Substitute: 3l² = 1, giving l = ±1/3\sqrt{3}

  5. Step 5: Direction cosines are (1/3\sqrt{3}, 1/3\sqrt{3}, 1/3\sqrt{3}) or (-1/3\sqrt{3}, -1/3\sqrt{3}, -1/3\sqrt{3})

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