Give the vector equation of the
line through the point three, seven, negative seven with direction vector zero,
negative five, seven.
We should remember that when we
need to write an equation in vector form, it will be in the form 𝐫 equals 𝐫 sub
zero plus 𝑡𝐯, where 𝐫 is the position vector of a general point on the line, 𝐫
sub zero is a position vector of a given point on the line, and 𝐯 is the direction
vector. 𝑡 is a scalar multiple. If we look at the information that
we’re given in the question, we can see that we have a direction vector. And we’ve got a point on the line
which can be written as a position vector. As we navigate from the origin to
the point three, seven, negative seven, then we can write this as the position
vector three, seven, negative seven.
We can then simply plug in these
two vectors into the vector form. 𝐫 equals the position vector
three, seven, negative seven plus 𝑡 times the direction vector zero, negative five,
seven. And so that’s the answer for the
vector equation of the line.