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

Explanation

Let `bar(a) "and" bar(b) ` be the position vectors of the points A and B w.r.t. The origin.
`therefore bar(a) = 3hati + 4hatj -7hatk "and" bar(b) = 6hati - hatj + hatk`
`therefore bar(b) - bar(a) = (6hati - hatj + hatk ) - ( 3hati + 4hatj - 7hatk )`
`= 3hati - 5hatj + 8hatk`
The vector equation of the line passing through the points A `bar(a) "and" B bar(b) ` is ` bar(r) = bar(a) + lamda (bar(b) - bar (a))`
`therefore` vector equation of the required line is
`bar(r) = (3hati + 4hatj - 7hatk ) + lamda ( 3hati - 5hatj + 8hatk)`
The cartesian form of the equation of the line passing through `(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` the cartesian form of the required line is ` (x - 3)/( 6 - 3) = ( y - 4)/(-1 -4) = (z - (-7))/(1 - (-7))`
i.e., `(x - 3)/(3) = ( y -4)/(-5) = (z + 7 )/(8)`