Question

Let L be the tine passing through the point P=(−5,−1,4) with ditection vector d=[−4,4,5] ᵀ , and let T be the plane defined by −2x+5y+3z=−37. Find the point Q where L and T intersect. Q=(0,0,0)

Answer 1

The student needs to find the intersection point Q by solving for the parameter t in the equation of plane T using the parametric equations derived from the line L passing through point P with direction vector d.

Step-by-step explanation:

The student is asking for the intersection point Q between the line L, which passes through the point P=(-5,-1,4) with direction vector d=[-4,4,5]T, and the plane T defined by the equation -2x+5y+3z=-37. To find the intersection, first write the parametric equations for the line L using point P and direction vector d. This gives us:

• x = -5 - 4t
• y = -1 + 4t
• z = 4 + 5t

Next, we substitute these parametric equations into the equation of the plane T to find the value of the parameter t at which the line intersects the plane:

-2(-5 - 4t) + 5(-1 + 4t) + 3(4 + 5t) = -37

Solving for t, we can then plug this value back into the parametric equations to find the Cartesian coordinates of point Q, which is the intersection point. The initial assumption of Q=(0,0,0) does not align with the provided values and must be derived from the equations.