Angle Between Two Lines (Vector Form)

Understanding the angle between two lines in three-dimensional space is a fundamental concept in vector algebra and geometry. When dealing with lines in vector form, we can determine the angle between them using the dot product of their direction vectors.

Basics of Vector Algebra

Before diving into the angle between two lines, let's review some basic vector algebra concepts:

• Vector: A quantity that has both magnitude and direction.
• Direction Vector: A vector that represents the direction of a line.
• Dot Product: An operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. Algebraically, it is the sum of the products of the corresponding entries of the two sequences of numbers.

Dot Product and the Angle Between Vectors

The dot product of two vectors $\vec{a}$ and $\vec{b}$ is defined as:

$$ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta) $$

where $|\vec{a}|$ and $|\vec{b}|$ are the magnitudes of vectors $\vec{a}$ and $\vec{b}$, respectively, and $\theta$ is the angle between them.

Formula for the Angle Between Two Lines

Given two lines with direction vectors $\vec{d_1}$ and $\vec{d_2}$, the cosine of the angle $\theta$ between the lines is given by:

$$ \cos(\theta) = \frac{\vec{d_1} \cdot \vec{d_2}}{|\vec{d_1}| |\vec{d_2}|} $$

To find the angle $\theta$, we take the inverse cosine (arccos) of the right-hand side:

$$ \theta = \arccos\left(\frac{\vec{d_1} \cdot \vec{d_2}}{|\vec{d_1}| |\vec{d_2}|}\right) $$

Important Points and Differences

Point	Description
Direction Vectors	The vectors that represent the direction of the lines.
Dot Product	A scalar that gives a measure of the vectors' parallelism.
Angle	The measure of the smallest rotation from one vector to another.

Examples

Example 1: Angle Between Two Perpendicular Lines

Consider two lines with direction vectors $\vec{d_1} = \langle 1, 0, 0 \rangle$ and $\vec{d_2} = \langle 0, 1, 0 \rangle$. These vectors are perpendicular to each other, so their dot product is zero:

$$ \vec{d_1} \cdot \vec{d_2} = (1)(0) + (0)(1) + (0)(0) = 0 $$

Thus, the cosine of the angle between them is zero, and the angle $\theta$ is:

$$ \theta = \arccos(0) = \frac{\pi}{2} \text{ or } 90^\circ $$

Example 2: Angle Between Two Skew Lines

Suppose we have two lines with direction vectors $\vec{d_1} = \langle 1, 2, 3 \rangle$ and $\vec{d_2} = \langle 4, -1, 2 \rangle$. To find the angle between them, we first calculate the dot product:

$$ \vec{d_1} \cdot \vec{d_2} = (1)(4) + (2)(-1) + (3)(2) = 4 - 2 + 6 = 8 $$

Next, we find the magnitudes of the direction vectors:

$$ |\vec{d_1}| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{14} $$

$$ |\vec{d_2}| = \sqrt{4^2 + (-1)^2 + 2^2} = \sqrt{21} $$

Now we can find the cosine of the angle:

$$ \cos(\theta) = \frac{8}{\sqrt{14} \sqrt{21}} $$

Finally, we find the angle $\theta$:

$$ \theta = \arccos\left(\frac{8}{\sqrt{14} \sqrt{21}}\right) $$

This angle can be evaluated using a calculator to get the angle in degrees or radians.

Conclusion

The angle between two lines in vector form can be determined using the dot product of their direction vectors. This concept is widely used in various fields such as physics, engineering, and computer graphics. Understanding how to calculate this angle is essential for solving geometric problems in three-dimensional space.