Answer to Question #188589 in Differential Equations for Johnny

Question #188589

Let L1 be the line in R3 with equation (x,y,z)=(1,0,2)+t(−1,3,4); t∈R

and let L2 be the line that is parallel to L1 and contains the point (1, −1, 3). Let V be the plane that contains both the lines L1 and L2.

(a) Find two vectors that are both parallel to the plane V but are not parallel to one another.

(b) Find a vector that is perpendicular to the plane V .

(c) Find an equation for the plane V .

(d) Find an equation for the line L3 that is perpendicular to the plane V and contains the point (1, −1, 4) .

Hint: Find a parametric equation for L3. Don’t try to find a Cartesian equation for L3.


1
Expert's answer
2021-05-07T11:39:15-0400

"L_1 : (x,y,z)=(1,0,2)+t(-1,3,4), t\\in R\\\\\nL_1 ||L_2 \\\\"


"L_1 : (x,y,z)=(1,-1,3)+t(-1,3,4), t\\in R\\\\\nV: L_1\\in V, L_2\\in V"


a)

"\\vec{a}=(-1,3,4)||V\\\\\nA(1,0,2)\\in L_1\\\\\nB(1,-1,3)\\in L_2\\\\"


"\\vec{AB}=(1-1,-1-0,3-2)=(0,-1,1)|| V\\\\\n\\vec{a}\\not|| \\vec{AB}\\\\\n\\frac{-1}{0}\\not=\\frac{3}{-1}\\not=\\frac{4}{1}"


b)

"\\vec{n}=[\\vec{a},\\vec{AB}]=\\begin{vmatrix}\n \\vec{i} & \\vec{j}&\\vec{k} \\\\\n -1&3&4\\\\\n0&-1&1 \n\\end{vmatrix}=\\\\\n=\\vec{i}(3+4)-\\vec{j}(-1-0)+\\vec{k}(1-0)=(7,1,1)"

"\\vec{n}" is perpendicular to the plane "V"



c)

"\\vec{n}=(7,1,1)" is perpendicular to the plane, "A(1,0,2)\\in V"

"7(x-1)+1(y-0)+1(z-2)=0\\\\\n7x+y+z-9=0"



d)

"L_3" is perpendicular to the plane "V"

"\\vec{n}=(7,1,1)||L_3\\\\\nC(1,-1,4)\\in L_3\\\\\n(x,y,z)=(1,-1,4)+t(7,1,1), t\\in R"



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