Question

Let L ₁ be the line passing through the points Q ₁ =(−5,5,−4) and Q ₂ =(−8,11,2) Find a value of k so the line L ₂ passing through the point P ₁ =P₁ (−6,−2,k) with direction vector d=[−3,3,−3] ᵀ intersects with L ₁ . k=0

Answer 1

To find k for the intersection of L₂ with L₁, we use the direction vectors and a system of equations derived from the parametric equations of both lines. Since k is given as 0, this is substituted directly into L₂'s equation to check for intersection.

Step-by-step explanation:

The student is asking to find a value for k so that line L₂, which passes through a point with coordinates P₁ (−6,−2,k), intersects with another line L₁. The direction vector for L₂ is given as d=[−3,3,−3]. To solve this, we need to use the fact that if the lines intersect, there exists a common point through which both lines pass.

We find the direction vector of line L₁ by subtracting coordinates of Q₁ from Q₂, resulting in the direction vector [−3,6,6]. Since lines intersect, let the parametric equations for L₁ and L₂ be equal on t and s respectively. This yields a system of equations. Solving this system gives the value of k when the lines intersect. However, given that we know k=0, we can directly substitute k into the equation of L₂ to obtain the specific point which must satisfy the equation of L₁.