The line passing through the points (2,8,10) and (3,11,8) is parallel to the line x=7+5t, y=2+15t, z=1-10t. true or false

To determine whether the line passing through the points \((2, 8, 10)\) and \((3, 11, 8)\) is parallel to the line given by the parametric equations:
\[
x = 7 + 5t, \quad y = 2 + 15t, \quad z = 1 - 10t,
\]
we need to compare the direction vectors of both lines.

1. **Direction vector of the line through the points \((2, 8, 10)\) and \((3, 11, 8)\):**
   The direction vector can be found by subtracting the coordinates of the two points:
   \[
   \mathbf{v_1} = (3 - 2, 11 - 8, 8 - 10) = (1, 3, -2).
   \]

2. **Direction vector of the parametric line:**
   The parametric line has the direction vector given by the coefficients of \(t\), which is:
   \[
   \mathbf{v_2} = (5, 15, -10).
   \]

3. **Check for parallelism:**
   Two vectors are parallel if one is a scalar multiple of the other. We check if there exists a constant \(k\) such that:
   \[
   (1, 3, -2) = k(5, 15, -10).
   \]
   Comparing the components:
   \[
   1 = 5k, \quad 3 = 15k, \quad -2 = -10k.
   \]
   Solving each equation:
   - From \(1 = 5k\), we get \(k = \frac{1}{5}\).
   - From \(3 = 15k\), we get \(k = \frac{1}{5}\).
   - From \(-2 = -10k\), we get \(k = \frac{1}{5}\).

Since the value of \(k\) is consistent across all three components, the direction vectors are scalar multiples of each other, which means the two lines are parallel.

Thus, the statement is **true**.

Would you like further details on this or have any questions?

Here are some related questions:
1. How do you find the parametric equation of a line passing through two points?
2. How can you check if two lines in 3D space intersect?
3. What is the condition for two lines to be perpendicular in 3D?
4. How do you calculate the angle between two vectors in 3D?
5. Can lines be both parallel and coincident in 3D space?

**Tip:** When checking for parallel lines, always compare the direction vectors to see if one is a scalar multiple of the other.

The line passing through the points (2,8,10) and (3,11,8) is parallel to the line x=7+5t, y=2+15t, z=1-10t. True or false?

Learn how to determine if the line passing through points (2,8,10) and (3,11,8) is parallel to the parametric line given by x=7+5t, y=2+15t, z=1-10t. This problem explores vector algebra and 3D geometry, including how to calculate direction vectors and check parallelism in space. Suitable for students in Grades 11-12.

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