Direction Cosines

Question

If l, m, n are direction cosines of a line, then l² + m² + n² equals

RankGuru · Accepted Answer

1. **Definition**: Direction cosines $l, m, n$ are the cosines of the angles $\alpha, \beta, \gamma$ that a line makes with the positive $x, y, z$ axes respectively: $l = \cos\alpha, m = \cos\beta, n = \cos\gamma$.
2. **Vector Analysis**: If $\vec{r}$ is a unit vector along the line, it can be expressed as $\vec{r} = l\hat{i} + m\hat{j} + n\hat{k}$.
3. **Magnitude**: Since $|\vec{r}| = 1$, we have $\sqrt{l^2 + m^2 + n^2} = 1$.
4. **Result**: Squaring both sides gives $l^2 + m^2 + n^2 = 1$. This is a fundamental property of direction cosines.

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