Question

If \((l_1, m_1, n_1), (l_2, m_2, n_2)\) are the direction cosines of two lines, then \[ (l_2m_1 - l_1m_2)^2 + (m_1n_2 - m_2n_1)^2 + (n_1l_2 - n_2l_1)^2 + (l_1l_2 + m_1m_2 + n_1n_2)^2 = \]

• A. \(0\)
• B. \(1\)
• C. \(2\)
• D. \(4\)

Hint:

Always recognize dot and cross product identities: \( |\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2 = |\vec{a}|^2|\vec{b}|^2 \).

Answer 1

The Correct Option is B

We know that the expression: \[ (l_2m_1 - l_1m_2)^2 + (m_1n_2 - m_2n_1)^2 + (n_1l_2 - n_2l_1)^2 \] is the square of the sine of the angle between two vectors \( \vec{a} \) and \( \vec{b} \), multiplied by the square of the magnitudes of those vectors. And: \[ (l_1l_2 + m_1m_2 + n_1n_2)^2 \] is the square of the cosine of the angle between them. So this sum becomes: \[ \sin^2 \theta + \cos^2 \theta = 1 \]