Find the vector wuation of the line passing through the points A(3,4, -7) and B (6,-1, 1)

Explanation

Let `bara` and `barb` be the position vector of the points A and b .
Then , `bara = 3hati + 4hatj - 7hatk`
and `" "barb = 6hati - hatj + hatk`
`barb - bara = (6hati - hatj + hatk) - (3hati + 4hatj - 7hatk)`
` = 3hati - 5hatj + 8hatk`
Now , vector equation of a line passing through A`(bara)` and `(B)(barb)` is
`barr = bara + lambda(barb - bara)`
` therefore" "barr = (3hati + 4hatk - 7hatk) + lambda(3hati - 5hatj + 8hatk)`
and the Cartesian equation of a line passing throught `(x_(1), y_(1),z_(1))` and `(x_(2),y_(2),z_(2))` is
`(x-x_(1))/(x_(2) -x_(1)) = (y - y_(1))/(y_(2) - y_(1)) = (z-z_(1))/(z_(2)-z_(1))`
`therefore " "(x - 3)/(6-3) = (y-4)/(-1-4) = (z - (-7))/(1-(-7))`
`i.e., " "(x-3)/(3)=(y - 4)/(-5)=(x+7)/(8)`