Semana 1B Vectores rectas paralelas

Main ideas / lesson conveyed

• The video explains how to find a parameter (m) so that the line through two given points is parallel to another given line.
• Key concept: Two lines are parallel iff their direction vectors are proportional.

Methodology / steps (detailed)

1) Read the task

You are given:

• Point (A = (-1,\,3,\,4))
• Point (B = (m,\,1,\,2))

You must find (m) such that the line through (A) and (B) is parallel to line (R), defined by:

[ \frac{x-3}{4}=\frac{y-2}{2}=z ]

2) Compute the direction vector of the line through (A) and (B)

Direction vector:

[ \vec{AB} = \vec{B} - \vec{A} ]

Subtract coordinates:

• (m - (-1) = m+1)
• (1 - 3 = -2)
• (2 - 4 = -2)

So:

[ \vec{AB} = (m+1,\,-2,\,-2) ]

3) Identify the direction vector of line (R)

From the given symmetric form (as stated in the subtitles), the direction vector is:

[ \vec{r} = (4,\,1,\,1) ]

4) Apply the parallel condition

If the lines are parallel, then their direction vectors are proportional:

[ (m+1,\,-2,\,-2)\ \parallel\ (4,\,1,\,1) ]

5) Set up proportionality

Using component ratios (as done in the subtitles):

[ \frac{m+1}{4}=\frac{-2}{1}=\frac{-2}{1} ]

6) Solve for (m)

From:

[ \frac{m+1}{4}=-2 ]

[ m+1 = 4(-2) = -8 ] [ m = -8 - 1 = -9 ]

Conclusion

• The value of the parameter that makes the line through (A) and (B) parallel to line (R) is:

[ \boxed{m=-9} ]

Speakers / sources featured

• Paola Pancaldi (teacher of algebra)