Let L be the line with parametric equations
x = −5+2t
y = −7−3t
z = 9−2t
Find the vector equation for a line that passes through the point P=(7, 3, −5) and intersects L at a point that is distance 5 from the point Q=(−5, −7, 9). Note that there are two possible correct answers.
To find the vector equation for a line that passes through point P and intersects line L at a point that is distance 5 from point Q, we need to find a point on line L that is 5 units away from point Q.
The vector equation for a line passing through point P can be written as:
r = P + vt
where r is the position vector of any point on the line, P is the position vector of the given point P, v is a direction vector of the line, and t is a scalar parameter.
Step 1: Find a point on line L that is 5 units away from point Q.
To do this, we need to find a point on line L that is distance 5 from point Q. We can use the distance formula to find the distance between two points:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
In this case, the distance between the point on line L and point Q is given as 5. So we have:
5 = sqrt((-5 - (-5+2t))^2 + (-7 - (-7-3t))^2 + (9 - (9-2t))^2)
Simplifying this equation will give us t values that correspond to the points on line L that are 5 units away from point Q.
Step 2: Find the direction vector of line L.
To find the direction vector of line L, we can take the coefficients of t in the parametric equations of line L:
direction vector v = <2, -3, -2>
Step 3: Write the vector equation for the line passing through point P and intersecting line L at a point that is distance 5 from point Q.
Using the point P and the direction vector v, we can write the vector equation as:
r = <7, 3, -5> + t<2, -3, -2> (where t is the parameter)
This is one possible correct answer. To find the second correct answer, you can use the negative value of t obtained from Step 1:
r = <7, 3, -5> + (-t)<2, -3, -2> (where -t is the parameter)
These two equations represent two different lines passing through point P and intersecting line L at points that are distance 5 from point Q.