Find the line through (8, 1, −5) that intersects and is perpendicular to the line x = −1 + t, y = −3 + t, z = −1 + t. (hint: if (x0, y0, z0) is the point of intersection, find its coordinates. enter your answers as a comma-separated list of equations.)

Step-by-step explanation:

A line passing through the point (8, 1, −5) and perpendicular to the line x = −1 + t, y = −3 + t, z = −1 + t.

The parallel vector of given line is < 1, 1, 1 >.

Let the point of intersection is  (-1+t, -3+t, -1+t).

The required line passing through the points (8, 1, −5) and  (-1+t, -3+t, -1+t). So, the vector of required line is

< 8-(-1+t), 1-(-3+t), -5-(-1+t) >

< 9-t, 4-t, -4-t >

The dot product of two perpendicular vectors is 0.

Since < 9-t, 4-t, -4-t > and < 1, 1, 1 > are perpendicular vectors, therefore the dot product of these vectors is 0.

The value of t is 3. So, the point of intersection is

If a line passes through two points then the equation of line is

The line passes through two points (8, 1, −5) and (2,0,2). So, the equation of line is

Let

Therefore the parametric equations of the required line are .