Question 17 of 68intermediate🔧 ApplyShort Answer2 marks

Show that the vector î + ĵ + k̂ is equally inclined to the axes OX, OY and OZ.

Correct Answer

Let the given vector be a = î + ĵ + k̂.

The direction cosines of a vector are found by dividing each component by the magnitude of the vector. The magnitude of the given vector is |a| = 12+12+12\sqrt{1² + 1² + 1²} = 3\sqrt{3}.

Therefore, the direction cosines are: l = 1/3\sqrt{3}, m = 1/3\sqrt{3}, n = 1/3\sqrt{3}.

Since all three direction cosines are equal (l = m = n = 1/3\sqrt{3}), the angles made with all three axes are equal. If α, β, γ are the angles made with OX, OY, OZ respectively, then cos(α) = cos(β) = cos(γ) = 1/3\sqrt{3}.

Hence, α = β = γ, proving that the vector î + ĵ + k̂ is equally inclined to the axes OX, OY and OZ.

Exercise: EXERCISE 10.2 | Q: 14 | (Page 17)
For More Understanding

Explanation

This question tests the concept of direction cosines. A vector is equally inclined to the coordinate axes when its direction cosines (l, m, n) are all equal. The direction cosines represent the cosines of angles the vector makes with x, y, z axes respectively. For vector î + ĵ + k̂, all components are 1, so after dividing by magnitude 3\sqrt{3}, we get equal direction cosines, proving equal inclination.

Solution Steps

  1. Step 1: Find magnitude of vector î + ĵ + k̂ = 12+12+12\sqrt{1² + 1² + 1²} = 3\sqrt{3}

  2. Step 2: Calculate direction cosines: l = 1/3\sqrt{3}, m = 1/3\sqrt{3}, n = 1/3\sqrt{3}

  3. Step 3: Since l = m = n, conclude that angles with all three axes are equal

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