Angle Between Two Lines | JEE Main Vector & 3D Geometry Crash Course

Angle Between Two Lines: A JEE Essential

In the realm of 3D Geometry, understanding the angle between two lines is super important for JEE Main. It pops up in various problems, mixing with vectors, planes, and more. So, let's dive deep and nail this concept!

Intersecting Lines and the Angle

Imagine two lines in 3D space. If they intersect, they form an angle. But how do we find it? We use direction cosines (DCs) and direction ratios (DRs) to make our lives easier.

Direction Cosines (DCs) to the Rescue

Direction cosines are the cosines of the angles that a line makes with the positive directions of the $x$ , $y$ , and $z$ axes. Represent them as $l$ , $m$ , and $n$ .

Formula 1: Angle using Direction Cosines

cos θ = |l₁l₂ + m₁m₂ + n₁n₂|

Explanation:

Let's say we have two lines with DCs $l₁, m₁, n₁$ and $l₂, m₂, n₂$ . The angle $θ$ between them is found using the formula above. This formula stems from the dot product of unit vectors along the two lines. Remember, the dot product $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}| cos θ$ . If $\vec{a}$ and $\vec{b}$ are unit vectors, then $\vec{a} \cdot \vec{b} = cos θ$ .

For example, if line 1 has DCs (1/√2, 1/√2, 0) and line 2 has DCs (1, 0, 0), then $cos θ = |(1/√2)(1) + (1/√2)(0) + (0)(0)| = 1/√2$ . Thus, $θ = 45°$ .

Direction Ratios (DRs) to the Rescue

Direction ratios are numbers proportional to the direction cosines. If $l, m, n$ are DCs, then $al, am, an$ are DRs, where $a$ is any non-zero number. Represent them as $a$ , $b$ , and $c$ .

Formula 2: Angle using Direction Ratios

cos θ = \frac{|a₁a₂ + b₁b₂ + c₁c₂|}{\sqrt{(a₁²+b₁²+c₁²)} \cdot \sqrt{(a₂²+b₂²+c₂²)}}

Explanation:

When you have the DRs $a₁, b₁, c₁$ and $a₂, b₂, c₂$ of two lines, this formula is your friend. It is derived from the same dot product principle as the DCs formula, but adjusted to account for the proportionality of DRs. We normalize the DRs to effectively convert them into DCs before applying the dot product.

For example, if line 1 has DRs (1, 1, 0) and line 2 has DRs (1, 0, 0), then

$cos θ = \frac{|(1)(1) + (1)(0) + (0)(0)|}{\sqrt{(1²+1²+0²)} \cdot \sqrt{(1²+0²+0²)}} = \frac{1}{\sqrt{2} \cdot 1} = \frac{1}{\sqrt{2}}$ . Thus, $θ = 45°$ .

Parallel Lines: The Ultimate Alignment

Two lines are parallel if their direction vectors are proportional.

Formula 3: Condition for Parallel Lines

\frac{a₁}{a₂} = \frac{b₁}{b₂} = \frac{c₁}{c₂} \text{ (or } l₁=l₂, m₁=m₂, n₁=n₂)

Explanation:

This condition ensures that the direction ratios (or DCs) of the two lines are in constant proportion. It means the lines are pointing in the same (or exactly opposite) direction.

For instance, lines with DRs (1, 2, 3) and (2, 4, 6) are parallel because 1/2 = 2/4 = 3/6.

Perpendicular Lines: Right Angles All Around

Two lines are perpendicular if the angle between them is 90°.

Formula 4: Condition for Perpendicular Lines

a₁a₂ + b₁b₂ + c₁c₂ = 0 \text{ (or } l₁l₂ + m₁m₂ + n₁n₂ = 0)

Explanation:

This condition arises from the fact that $cos 90° = 0$ . So, the dot product of their direction vectors must be zero.

Example: Lines with DRs (1, 1, 0) and (0, 0, 1) are perpendicular because (1)(0) + (1)(0) + (0)(1) = 0.

Tip: Always remember to take the modulus (absolute value) in the angle formulas. This ensures you're finding the acute angle between the lines.

Common Mistake: Forgetting to normalize DRs before using them in the angle formula. DRs need to be converted to DCs (by dividing by the magnitude of the direction vector) if you want to use the

l₁l₂ + m₁m₂ + n₁n₂ = 0

form.

JEE Trick: If a question involves finding the angle between a line and a plane, first find the angle between the line and the normal vector to the plane, and then subtract it from 90° to get the required angle.

Mastering these concepts and formulas will give you a solid edge in tackling 3D geometry problems in JEE Main. Keep practicing, and you'll ace it!