Find the vector equation of the line passing through the point `(1," "2," "" "4)` and perpendicular to the two lines: `(x-8)/3=(y+19)/(-16)=(z-10)/7` and `(x-15)/3=(y-29)/8=(z-5)/(-5)`

Explanation

Since the line to be determined is perpendicular to the given two straight lines, it is directed towards
`" ""Vector "(3hati-16hatj+7hatk)xx(3hati+8hatj-5hatk)` `=|{:(hati,,hatj,,hatk),(3,,-16,,7),(3,,8,,-5):}|=24hati+36hatj+72hatk`
Therefore, direction ratios of the line are 24, 36, 72 or 2, 3, 6.
Hence, equation of line passing through the point (1, 2, -4) and parallel to the vector `2hati+3hatj+6hatk` is
`" "vecr=(hati+2hatj-4hatk)+lamda(2hati+3hatj+6hatk)`
This is the equation of the required line.