Determine whether the lines are parallel or identical, x-1/4=y-1/2=z+3/4, x+2/1=y-1/0.5=z-3/1

We are given two lines in symmetric form, and we need to determine whether the lines are **parallel** or **identical**. Let's begin by converting the symmetric form of each line into vector form to analyze them more clearly.

### Line 1: 
\[
\frac{x - 1}{4} = \frac{y - 1}{2} = \frac{z + 3}{4}
\]

This represents the line in symmetric form. We can rewrite this in parametric form by introducing a parameter \( t \):
\[
x = 1 + 4t, \quad y = 1 + 2t, \quad z = -3 + 4t
\]
Thus, the direction vector for Line 1 is:
\[
\mathbf{d_1} = \langle 4, 2, 4 \rangle
\]

### Line 2:
\[
\frac{x + 2}{1} = \frac{y - 1}{0.5} = \frac{z - 3}{1}
\]

We can rewrite this in parametric form by introducing a parameter \( s \):
\[
x = -2 + s, \quad y = 1 + 0.5s, \quad z = 3 + s
\]
Thus, the direction vector for Line 2 is:
\[
\mathbf{d_2} = \langle 1, 0.5, 1 \rangle
\]

### Checking if the lines are parallel:
Two lines are parallel if their direction vectors are scalar multiples of each other, i.e., there exists a constant \( k \) such that:
\[
\mathbf{d_1} = k \mathbf{d_2}
\]
Let's check if this is the case for our direction vectors:
\[
\mathbf{d_1} = \langle 4, 2, 4 \rangle, \quad \mathbf{d_2} = \langle 1, 0.5, 1 \rangle
\]
We can divide each component of \( \mathbf{d_1} \) by the corresponding component of \( \mathbf{d_2} \):
\[
\frac{4}{1} = 4, \quad \frac{2}{0.5} = 4, \quad \frac{4}{1} = 4
\]
Since all ratios are equal to 4, the direction vectors are scalar multiples of each other. Therefore, the lines are **parallel**.

### Checking if the lines are identical:
To check if the lines are identical, we need to determine if there is a point that lies on both lines. To do this, we can equate the parametric equations of the two lines and see if a common solution exists for \( t \) and \( s \).

For Line 1: 
\[
x = 1 + 4t, \quad y = 1 + 2t, \quad z = -3 + 4t
\]

For Line 2:
\[
x = -2 + s, \quad y = 1 + 0.5s, \quad z = 3 + s
\]

Equating the coordinates of the two lines:

1. \( 1 + 4t = -2 + s \)
2. \( 1 + 2t = 1 + 0.5s \)
3. \( -3 + 4t = 3 + s \)

From equation (2):
\[
2t = 0.5s \quad \Rightarrow \quad s = 4t
\]

Substitute \( s = 4t \) into equations (1) and (3):

1. \( 1 + 4t = -2 + 4t \quad \Rightarrow \quad 1 = -2 \) (This is a contradiction)
3. \( -3 + 4t = 3 + 4t \quad \Rightarrow \quad -3 = 3 \) (This is also a contradiction)

Since we get contradictions, the lines do not have a common point and are therefore not identical.

### Conclusion:
The two lines are **parallel** but not identical.

Would you like further details or have any questions? Here are some related questions to consider:

1. How do you find the angle between two lines in 3D?
2. Can parallel lines in 3D have a common point?
3. How do you determine the shortest distance between two parallel lines?
4. What is the significance of direction vectors in line equations?
5. How can you check if two skew lines intersect in 3D?

**Tip:** To test if lines are parallel, always compare their direction vectors. If they are scalar multiples, the lines are parallel.

Analyze whether two 3D lines, given in symmetric form, are parallel or identical by comparing their direction vectors and checking for a common point. Explore vector algebra and line equations with this detailed explanation, suitable for students in Grades 10-12.

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