Question

1. (Sections 2.3, 2.10, 2.11, 2.12) Let L1 be the line in R 3 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. (2) (b) Find a vector that is perpendicular to the plane V . (2) (c) Find an equation for the plane V . (2) (d) Find an equation for the line L3 that is perpendicular to the plane V and contains the point (1, −1, 4) . (2) Hint: Find a parametric equation for L3. Don’t try to find a Cartesian equation for L3. (Study Remarks 2.12.2.)

Answer 1

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

a)

$\vec{a}=(-1,3,4)||V\\ A(1,0,2)\in L_1\\ B(1,-1,3)\in L_2\\ \vec{AB}=(1-1,-1-0,3-2)=(0,-1,1)|| V\\ \vec{a}\not|| \vec{AB}\\ \frac{-1}{0}\not=\frac{3}{-1}\not=\frac{4}{1}$

b)

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

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

c)

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

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

d)

$L_3$ is perpendicular to the plane $V$

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