If a line makes an angle of π/4 with the positive directions of each of x-axis and y-axis, then the

Q: Wait, why is the sum of the squares of direction cosines always 1? Is there a geometric proof?

Absolutely! Think of a unit vector $\vec{u}$ along the line. Its components are $(\cos \alpha, \cos \beta, \cos \gamma)$. Since it is a unit vector, its magnitude must be 1. By the definition of magnitude, $\sqrt{\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma} = 1$, which squares to our fundamental identity. It is the 3D version of the Pythagorean theorem!

Q: Can $\gamma$ be something other than $\pi/2$?

In the context of direction angles, we define $\gamma$ in the range $[0, \pi]$. Since $\cos \gamma = 0$ is the only solution in this interval, $\pi/2$ is the unique answer. It means the line is perfectly perpendicular to the z-axis.

Q: Does this line lie entirely in the xy-plane?

Yes! If the angle with the z-axis is $\pi/2$, the line has no 'vertical' component. It is parallel to the xy-plane. Since it passes through the origin, it must lie exactly within the xy-plane.

Q: What if the angles were different, say $\pi/3$ and $\pi/3$? Would the math still hold?

The beauty of the identity $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$ is its universality. As long as the angles correspond to a valid line in 3D space, the identity will always hold. If you chose angles that made the sum exceed 1, such a line simply could not exist!

Q: Is there any ambiguity about 'positive direction' of the axes?

The 'positive direction' is simply the standard orientation of the coordinate axes. By specifying this, we ensure we are looking for the principal angle $\gamma \in [0, \pi]$, keeping our calculations clean and unambiguous.

Question

If a line makes an angle of $\pi/4$ with the positive directions of each of $x$-axis and $y$-axis, then the angle that the line makes with the positive direction of the $z$-axis is

Options:

(A) $\pi/4$

(B) $\pi/2$

(C) $\pi/6$

(D) $\pi/3$

Sigma Percentile · Accepted Answer

### The Geometry of Orientation: A Journey into 3D Space

Imagine you are standing at the origin of a three-dimensional Cartesian coordinate system. You have the $x$-axis stretching to your right, the $y$-axis extending forward, and the $z$-axis pointing straight up toward the ceiling.

You are holding a long, thin rod, and you decide to tilt it such that it makes an angle of $\frac{\pi}{4}$ with both the $x$-axis and the $y$-axis. Have you ever wondered what this implies for the rod's relationship with the vertical $z$-axis?

Today, we are going to uncover the elegant mathematics that governs this orientation.

### The Language of Direction Cosines

In 3D geometry, we describe the orientation of a line using 'direction angles'—$\alpha, \beta,$ and $\gamma$—which represent the angles the line makes with the positive $x, y,$ and $z$ axes, respectively. These angles are the keys to unlocking the line's identity.

We define the direction cosines as $l = \cos \alpha$, $m = \cos \beta$, and $n = \cos \gamma$.

There is a profound, fundamental identity that binds these three values together:

$$ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 $$

Think of this as the 3D extension of the Pythagorean theorem. Just as $x^2 + y^2 = r^2$ defines a circle in 2D, this identity defines the constraint on the orientation of any line in 3D space. It tells us that the 'total influence' of the line's direction across all three axes must be perfectly balanced.

### The Calculation: Unveiling the Mystery

We are given that our rod makes an angle of $\frac{\pi}{4}$ with the $x$-axis and the $y$-axis. Thus, $\alpha = \frac{\pi}{4}$ and $\beta = \frac{\pi}{4}$.

Our goal is to find $\gamma$. Let us substitute these values into our identity:

$$ \cos^2\left(\frac{\pi}{4}\right) + \cos^2\left(\frac{\pi}{4}\right) + \cos^2 \gamma = 1 $$

We know from our trigonometric foundations that $\cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$. When we square this value, we get:

$$ \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} $$

Substituting this back into our equation, the magic begins to happen:

$$ \frac{1}{2} + \frac{1}{2} + \cos^2 \gamma = 1 $$

Adding the fractions, we see that $\frac{1}{2} + \frac{1}{2} = 1$. The equation simplifies beautifully to:

$$ 1 + \cos^2 \gamma = 1 $$

### The Final Revelation

Subtracting $1$ from both sides, we arrive at $\cos^2 \gamma = 0$, which implies $\cos \gamma = 0$. In the interval $[0, \pi]$, the only angle whose cosine is zero is **$\gamma = \frac{\pi}{2}$**.

What does this mean physically? It means that our rod is perfectly perpendicular to the $z$-axis.

Because it is perpendicular to the vertical axis, it must lie entirely within the horizontal $xy$-plane. By simply knowing the angles with two axes, the third angle was forced into existence by the rigid laws of geometry.

Animated Solution for Mathematics - Three Dimensional Geometry: If a line makes an angle of $π /4$ with the positive directions of each of $x$ -axis and $y$ -axis, then the angle that the line makes with the positive direction of the $z$ -axis is

Select Answer:

Visualized Solution

The 3D Coordinate System

Direction Angles

Given Angles

The Fundamental Identity

Substituting Known Values

Evaluating $cos (\frac{π}{4})$

Squaring the Terms

Simplifying the Equation

Solving for $cos γ$

Finding the Final Angle

The Sigma Insight: Direction Cosines and Direction Ratios

The Geometry of Orientation

The Language of Direction Cosines

The Calculation

The Final Revelation

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