The angle between the lines 2x = 3y = -z and 6x = -y = -4z is
(a) 0°
(b) 90°
(c) 45°
(d) 30°

Q: Wait, why do we divide by the LCM? Can't I just pick any numbers that satisfy the ratio?

That is a fantastic question! You are absolutely right that any set of numbers proportional to the direction ratios will define the same line. However, dividing by the LCM is a systematic way to ensure you get the *standard* direction ratios directly from the equation. It removes the guesswork and prevents errors when you plug those values into the angle formula. Think of it as normalizing the equation so the geometry becomes transparent.

Q: Why do we use the absolute value in the numerator of the angle formula?

Great observation! The angle between two lines is conventionally defined as the acute angle, which is why we use the absolute value. If we didn't, we might get a negative cosine value, which would give us the obtuse angle. Since the lines are infinite, they intersect at two supplementary angles; we just choose the acute one by convention.

Q: What if the dot product wasn't zero? How do I handle the square roots in the denominator?

If the dot product isn't zero, you simply proceed with the calculation. The square roots in the denominator represent the magnitudes of the direction vectors. You calculate each magnitude separately, multiply them, and then divide the dot product by that result. It is just arithmetic—don't let the radicals intimidate you! Just keep your steps organized.

Question

The angle between the lines $2x = 3y = -z$ and $6x = -y = -4z$ is

Options:

(A) $0^\circ$

(B) $90^\circ$

(C) $45^\circ$

(D) $30^\circ$

Sigma Percentile · Accepted Answer

### The Geometry of Lines: Unmasking the 3D Mystery

Welcome, my dear student. Today, we are not just solving a problem; we are embarking on a journey into the heart of three-dimensional space.

Many students look at a problem like this—finding the angle between two lines—and see only a mess of variables. But I want you to see the elegance. I want you to see the hidden geometry.

### The Disguise: Why $2x = 3y = -z$ is Tricky

Look at the equations provided: $2x = 3y = -z$ and $6x = -y = -4z$. At first glance, they look like simple algebraic expressions. But they are in disguise.

In the world of 3D geometry, a line is most powerful when it is in its 'symmetric form':

$$\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$$

In this form, the denominators $(a, b, c)$ are the 'DNA' of the line—the direction ratios. Our first task is to unmask these lines.

### The Transformation: The LCM Method

To transform $2x = 3y = -z$ into the symmetric form, we need the coefficients of $x$, $y$, and $z$ to be exactly $1$. We look at the coefficients: $2$, $3$, and $-1$.

The least common multiple (LCM) of the magnitudes $2$ and $3$ is $6$. So, we divide the entire equation by $6$:

$$\frac{2x}{6} = \frac{3y}{6} = \frac{-z}{6} \Rightarrow \frac{x}{3} = \frac{y}{2} = \frac{z}{-6}$$

Now, the line is unmasked! The direction ratios are $(3, 2, -6)$.

We repeat this for the second line: $6x = -y = -4z$. The coefficients are $6$, $-1$, and $-4$. The LCM of $6$ and $4$ is $12$.

Dividing by $12$, we get:

$$\frac{6x}{12} = \frac{-y}{12} = \frac{-4z}{12} \Rightarrow \frac{x}{2} = \frac{y}{-12} = \frac{z}{-3}$$

Now we have our second set of direction ratios: $(2, -12, -3)$.

### The Collision: The Dot Product

Now, we have two vectors representing the directions of our lines: $\vec{b}_1 = 3\hat{i} + 2\hat{j} - 6\hat{k}$ and $\vec{b}_2 = 2\hat{i} - 12\hat{j} - 3\hat{k}$. To find the angle $	heta$ between them, we use the dot product formula:

$$\cos 	heta = \frac{|a_1 a_2 + b_1 b_2 + c_1 c_2|}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}}$$

Let's calculate the numerator, the dot product: $(3)(2) + (2)(-12) + (-6)(-3)$.

This simplifies to $6 - 24 + 18$. Look at that! $6 + 18 = 24$, and $24 - 24 = 0$. The numerator is zero!

### The Revelation: Perpendicularity

When the dot product of two direction vectors is zero, it is a geometric signature. It means the vectors are orthogonal, and the lines are perfectly perpendicular.

Since $\cos 	heta = 0$, we know that **$	heta = 90^\circ$**.

We didn't even need to calculate the magnitudes in the denominator because zero divided by anything (non-zero) is still zero. This is the beauty of the JEE—sometimes, the math collapses into a simple, elegant truth.

Keep this in your toolkit: whenever you see two lines in 3D, find their direction ratios first. If their dot product is zero, you have found a perpendicular intersection. You have mastered the geometry.

Animated Solution for Mathematics - Three Dimensional Geometry: The angle between the lines $2 x = 3 y = - z$ and $6 x = - y = - 4 z$ is

Select Answer:

Visualized Solution

Visualizing the Lines in 3D Space

The Symmetric Form of a Line

Converting Line $L_{1}$ to Symmetric Form

Extracting Direction Ratios for $L_{1}$

Converting Line $L_{2}$ to Symmetric Form

Extracting Direction Ratios for $L_{2}$

The Angle Formula

Calculating the Dot Product

Final Result: Perpendicular Lines

Key Takeaways and Next Challenges

The Sigma Insight: Angle Between Two Lines

The Geometry of Lines

The Disguise

The Transformation

The Collision

The Revelation

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