Question

Is the line passing through (-4,-6,1) and (-2,0,-3) parallel to the line passing through (10,18,4) and (5,3,14)?

1) True
2) False

Answer 1

The given statement “Is the line passing through (-4,-6,1) and (-2,0,-3) parallel to the line passing through (10,18,4) and (5,3,14)?” is false because the lines passing through (-4, -6, 1) and (-2, 0, -3), and (10, 18, 4) and (5, 3, 14) are not parallel.Thus,the correct option is 2.

Step-by-step explanation:

The direction vector
`\(\mathbf{d_1}\)` for the line passing through (-4, -6, 1) and (-2, 0, -3) can be found by subtracting the coordinates of the two points:

`\[\mathbf{d_1} = \begin{bmatrix} -2 - (-4) \\ 0 - (-6) \\ -3 - 1 \end{bmatrix} = \begin{bmatrix} 2 \\ 6 \\ -4 \end{bmatrix}\]`

Similarly, the direction vector
`\(\mathbf{d_2}\)` for the line passing through (10, 18, 4) and (5, 3, 14) is:

`\[\mathbf{d_2} = \begin{bmatrix} 5 - 10 \\ 3 - 18 \\ 14 - 4 \end{bmatrix} = \begin{bmatrix} -5 \\ -15 \\ 10 \end{bmatrix}\]`

Two lines are parallel if their direction vectors are proportional. In other words,
`\(\mathbf{d_1} = k \cdot \mathbf{d_2}\)` for some constant (k). However, in this case, it is evident that
`\(\mathbf{d_1}\) and \(\mathbf{d_2}\)` are not proportional, as their scalar multiples do not match. Therefore, the lines are not parallel.

In conclusion, the line passing through (-4, -6, 1) and (-2, 0, -3) is not parallel to the line passing through (10, 18, 4) and (5, 3, 14). The direction vectors for the two lines are not scalar multiples of each other, confirming that they do not share the same direction and, therefore, are not parallel.

Therefore,the correct option is 2.