Finding the Angle between Two Lines

To find the angle between two lines, we need to determine the direction vectors of the lines and then apply the formula for finding the angle between two vectors. Let's break down the process step by step:

Step 1: Determine the Direction Vectors of the Lines
To find the direction vectors of the lines, we look at the coefficients of the variables i, j, and k in the given equations. The direction vector of a line is the vector that describes the direction in which the line is moving.

Given:
Line α: r = 3i + 2j - 4k
Line β: r = 5i - 2k

The direction vector of Line α is <3, 2,="" -4="">.
The direction vector of Line β is <5, 0,="" -2="">.

Step 2: Apply the Formula for Finding the Angle between Two Vectors
The formula to find the angle θ between two vectors A and B is:

cosθ = (A • B) / (|A| * |B|)

where • represents the dot product and |A| and |B| represent the magnitudes of vectors A and B, respectively.

Step 3: Calculate the Dot Product and Magnitudes
Let's calculate the dot product and magnitudes using the given direction vectors:

Dot product of Line α and Line β:
A • B = (3 * 5) + (2 * 0) + (-4 * -2) = 15 + 0 + 8 = 23

Magnitude of Line α:
|A| = sqrt(3^2 + 2^2 + (-4)^2) = sqrt(9 + 4 + 16) = sqrt(29)

Magnitude of Line β:
|B| = sqrt(5^2 + 0^2 + (-2)^2) = sqrt(25 + 0 + 4) = sqrt(29)

Step 4: Calculate the Angle
Using the formula for finding the angle between two vectors, we can now calculate the angle θ:

cosθ = (A • B) / (|A| * |B|)
cosθ = 23 / (sqrt(29) * sqrt(29))
cosθ = 23 / 29
θ = arccos(23 / 29)

Using a calculator, we find that arccos(23 / 29) is approximately 0.733 radians or 42.05 degrees.

Step 5: Final Answer
The angle between Line α and Line β is approximately 0.733 radians or 42.05 degrees.