Let L be the line with parametric equations x=5+t, y=6,z=−2−3t. Find the vector equation for a line that passes through the point P=(−5,7,−8) and intersects L at a point that is distance 3 from the point Q=(5,6,−2). Note that there are two possible correct answers. Use the square root symbol √ where needed to give an exact value for your answer.

The required equations are

and

.

Step-by-step explanation:

The given parametric equation of the line, , is so, an arbitrary point on the line is

The vector equation of the line passing through the points and is

The given equation can also be written as

The standard  equation of the line passes through the point and having direction is

Here, The value of the parameter,, of any point at a distance from the point, , can be determined by

Comparing equations and

The line is passing through the point having direction .

Note that the point is the same as obtained above.

Now, the value of the parameter, , for point at distance from the point can be determined by equation , we have

Put the value of in equation , the required equations are as follows:

For

and for ,

you need to show the graph

step-by-step explanation: