Q.Find the angle between the lines whose direction cosines are given by the equations $l + m + n = 0$, $l^2 + m^2 - n^2 = 0$.

Concept: Angle Between Lines — using direction cosines to find the angle between two lines.

Step 1: From $l + m + n = 0$, we have $n = -(l + m)$. Substitute into $l^2 + m^2 - n^2 = 0$:

So $lm = 0$, meaning either $l = 0$ or $m = 0$.

Step 2:

• If $l = 0$, then $m + n = 0 \implies n = -m$. Direction ratios: $(0, 1, -1)$.
• If $m = 0$, then $l + n = 0 \implies n = -l$. Direction ratios: $(1, 0, -1)$. …

Why This Approach Works

When two lines are defined by direction cosines $(l, m, n)$ that satisfy given equations, we are essentially finding the intersection of two surfaces in direction-cosine space. Each equation restricts the possible directions; solving them together gives us the actual direction vectors of the lines. The angle between the lines is then simply the angle between these vectors.

The first equation $l + m + n = 0$ is a plane through the origin. The second $l^2 + m^2 - n^2 = 0$ is a cone. Their intersection yields two distinct lines through the origin — exactly the two lines we need.

Step-by-Step Solution

1. Express one variable in terms of the others

From $l + m + n = 0$, we have:

2. Substitute into the second equation

3. Case 1: $l = 0$

With $l = 0$, the first equation gives $0 + m + n = 0$, so $n = -m$.

The direction ratios for this line are $(0, m, -m)$, or simply $(0, 1, -1)$ after scaling.

4. Case 2: $m = 0$

With $m = 0$, the first equation gives $l + 0 + n = 0$, so $n = -l$.

The direction ratios for this line are $(l, 0, -l)$, or simply $(1, 0, -1)$ after scaling. …

Method: Angle between two lines given by direction-cosine equations

Use this when the two lines are not given directly, but their direction cosines $(l,m,n)$ are described by two equations (typically one linear, one homogeneous quadratic).

Steps

Step 1: Eliminate one variable using the linear equation.

Solve the linear relation for one letter (say $n$) and substitute into the quadratic. You are left with a single homogeneous relation between the other two.

Step 2: Factor to get the two directions.

The reduced relation factors (or collapses to a simple condition such as a product being zero). Each factor or case gives the direction ratios of one of the two lines; scale to convenient integers. …

Common Mistakes

Mistake 1: Solving for the actual direction cosines with $l^2+m^2+n^2=1$.

Why it's wrong: it adds needless algebra — the angle formula works with direction ratios since the magnitudes cancel. Correct approach: from $lm=0$ just take ratios $(0,1,-1)$ and $(1,0,-1)$ and substitute.

Mistake 2: Losing one of the two cases from $lm=0$.

Why it's wrong: $lm=0$ means $l=0$ or $m=0$; each gives one line, and you need both to form the angle. Correct approach: treat the two cases separately, then take the angle between the two resulting directions. …