3D Geometry
Direction Cosines and Direction Ratios
Grade 12
Question:
<p>If <i>P</i> be (<i>x</i>, <i>y</i>, <i>z</i>), then from the figure <i>x</i> = <i>r</i> sin θ cos φ, <i>y</i> = <i>r</i> sin θ sin φ and <i>z</i> = <i>r</i> cos θ. Given 1 = <i>r</i> sin θ cos φ, 2 = <i>r</i> sin θ sin φ and 3 = <i>r</i> cos θ. Which of the following are correct?</p>
<p>(a) \(\sin\theta\cos\phi = \dfrac{1}{\sqrt{14}}\)</p>
<p>(b) \(\sin\theta\sin\phi = \dfrac{2}{\sqrt{14}}\)</p>
<p>(c) \(\cos\theta = \dfrac{3}{\sqrt{14}}\)</p>
<p>(d) \(r = \pm\sqrt{14}\)</p>
Step-by-Step Solution
Key Concept: Convert Cartesian coordinates (1, 2, 3) to spherical coordinates using the transformation equations to find r, θ, and φ, then verify relationships and properties.
Step 1: Given x = 1, y = 2, z = 3 in Cartesian form, use the spherical coordinate relations. Step 2: Find r: r^2 = x^2 + y^2 + z^2 = 1 + 4 + 9 = 14, so r = √14 Step 3: Find θ using z = r cos θ: cos θ = 3/√14, so θ = cos⁻^1(3/√14) Step 4: Find φ using x = r sin θ cos φ and y = r sin θ sin φ. Note that sin θ = √(1 - 9/14) = √(5/14) Step 5: From the two equations: tan φ = y/x = 2/1 = 2, so φ = tan⁻^1(2) Step 6: Verify: r sin θ cos φ = √14 · √(5/14) · 1/√5 = 1 ✓ and r sin θ sin φ = √14 · √(5/14) · 2/√5 = 2 ✓ and r cos θ = √14 · 3/√14 = 3 ✓ ∴ Answer: A, B, C, D (all relationships are consistent with spherical coordinates)
Correct Answer: A, B, C, D