Direction Cosines from Direction Angles in 3D

We are given a line in 3D space that makes the following angles with the coordinate axes:

• With the x-axis: $\alpha = 90^\circ$
• With the y-axis: $\beta = 135^\circ$
• With the z-axis: $\gamma = 45^\circ$

The direction cosines of a line are defined as:

$$\ell = \cos\alpha, \quad m = \cos\beta, \quad n = \cos\gamma$$

These form the components of a unit vector in the direction of the line:

$$\mathbf{u} = (\ell, m, n)$$

and they must satisfy the identity

$$\ell^2 + m^2 + n^2 = 1$$

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Step 1: Compute each cosine

1. With the x-axis

$$\alpha = 90^\circ \Rightarrow \ell = \cos 90^\circ = 0$$

2. With the y-axis

$$\beta = 135^\circ \Rightarrow m = \cos 135^\circ = \cos(180^\circ - 45^\circ) = -\cos 45^\circ = -\frac{1}{\sqrt{2}}$$

3. With the z-axis

$$\gamma = 45^\circ \Rightarrow n = \cos 45^\circ = \frac{1}{\sqrt{2}}$$

So the direction cosines are:

$$(\ell, m, n) = \left(0, -\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$$

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Step 2: Verify the identity

Check that they form a unit vector:

$$\ell^2 + m^2 + n^2 = 0^2 + \left(-\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 = 0 + \frac{1}{2} + \frac{1}{2} = 1$$

The condition is satisfied, so the direction cosines are consistent.

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Final Answer

The direction cosines of the line are:

$$\boxed{\ell = 0, \quad m = -\frac{1}{\sqrt{2}}, \quad n = \frac{1}{\sqrt{2}}}$$

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Visualization Idea

The canvas shows the 3D axes (x, y, z projected in perspective) and a unit vector representing the line, with components equal to the direction cosines:

$$\mathbf{u} = (\ell, m, n) = (0, -\tfrac{1}{\sqrt{2}}, \tfrac{1}{\sqrt{2}})$$

You can interactively change the three angles $\alpha, \beta, \gamma$ with sliders. The visualization will:

• Compute $\ell = \cos \alpha$, $m = \cos \beta$, $n = \cos \gamma$
• Draw a 3D-ish coordinate frame
• Plot the direction vector with those cosines
• Draw projections of the vector onto each axis
• Show a small check of $\ell^2 + m^2 + n^2$ as a bar: if it is 1, the bar is neon cyan; if not, it fades towards pink to hint that the three angles are not consistent with a single line.

Use the toggle to snap to the problem's given angles (90°, 135°, 45°) and see the exact solution vector.