The vector equation of a line passing through two points is x=a+λd, where a is a position vector of any point on the line, d is the direction vector, and λ is a parameter.
The direction vector is found by subtracting the coordinates of the two points:
d=B−A
d=(1−3)i+(−1−4)j+(6−(−7))k
d=−2i−5j+13k
Using point A(3,4,−7) as the position vector:
x=(3i+4j−7k)+λ(−2i−5j+13k)
Expanding the equation:
x=3i+4j−7k−2λi−5λj+13λk
Grouping terms:
x=(3−2λ)i+(4−5λ)j+(−7+13λ)k
Therefore, the vector equation of the line is x=(3−2λ)i+(4−5λ)j+(−7+13λ)k