If the direction cosines <l, m, n> of a line are connected by relation l + 2m + n = 0, 2l – 2m + 3n = 0, then what is the value of l² + m² – n²?

From l + 2m + n = 0 and 2l – 2m + 3n = 0, add: 3l + 4n = 0, so l = –4n/3. From l + 2m + n = 0: 2m = –l – n = 4n/3 – n = n/3, so m = n/6. Direction ratios proportional to (–4n/3, n/6, n), or multiplying by 6: (–8, 1, 6). Magnitude = √(64 + 1 + 36) = √101. So l² = 64/101, m² = 1/101, n² = 36/101. Then l² + m² – n² = (64 + 1 – 36)/101 = 29/101.