Let L be the line r=(2,-2,-1)+lambda(-1,0,1), and P=(2,-1,1). Plane E contains A=(0,1,1), B=(1,2,2), C=(1,0,1). Find: (a) distance from P to L; (b) equation of E; (c) equation of plane F containing L

Question

Let L be the line r=(2,-2,-1)+lambda(-1,0,1), and P=(2,-1,1). Plane E contains A=(0,1,1), B=(1,2,2), C=(1,0,1). Find: (a) distance from P to L; (b) equation of E; (c) equation of plane F containing L and P; (d) acute angle between E and F; (e) volume of the parallelepiped with A,B,C,D=(-3,0,1).

TestHub · Accepted Answer

Use the point Q and direction d of L. The normals to E and F follow from the cross products below. The final scalar triple product gives the parallelepiped volume. Q = ( 2 , − 2 , − 1 ) , d = ( − 1 , 0 , 1 ) , P − Q = ( 0 , 1 , 2 ) . (a) h 2 = 5 − 2 2 2 ​ = 3 , h = 3 ​ . (b) A B × A C = ( 1 , 1 , − 2 ) , E : x + y − 2 z + 1 = 0. (c) d × ( P − Q ) = ( − 1 , 2 , − 1 ) , F : x − 2 y + z = 5. (d) cos θ = 6 ​ 6 ​ 3 ​ = 2 1 ​ , θ = 3 π ​ . (e) V = ∣ ( A B × A C ) ⋅ A D ∣ = ∣ − 4∣ = 4. ​

Mathematics - 3D-Coordinate Geometry Question with Solution | TestHub

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