Section II.2

§ 2

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Problem II.2.1

Find the parametric equations of the line given by the equations

x+y+z=4,x−y+3z=0. \begin{aligned} & x+y+z=4, \\ & x-y+3 z=0 . \end{aligned}
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Problem II.2.2

Find the parametric equations of the plane x−2y+3z=1x-2 y+3 z=1.

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Problem II.2.3

Find the coordinates of the point of intersection of the lines with the equations: x=1−t,y=1+t,z=1−tx=1-t, y=1+t, z=1-t and x=3t−1,y=2t−2x=3 t-1, y=2 t-2, z=1+tz=1+t. What value of the parameter corresponds to this point on each of the lines? How can one establish that the lines meet without finding the point of intersection?

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Problem II.2.4

Write the equation of the plane in which the lines of exercise 3 lie.

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Problem II.2.5

Write the parametric equations of the lines given by the vector equations

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(a)

[r,a]=b,(a,b)=0,a≠0[\mathbf{r}, \mathbf{a}]=\mathbf{b},(\mathbf{a}, \mathbf{b})=0, \mathbf{a} \neq \mathbf{0}.

(b)

(r,n1)+D1=0,(r,n2)+D2=0,[n1,n2]≠0\left(\mathbf{r}, \mathbf{n}_{1}\right)+D_{1}=0,\left(\mathbf{r}, \mathbf{n}_{2}\right)+D_{2}=0,\left[\mathbf{n}_{1}, \mathbf{n}_{2}\right] \neq \mathbf{0}.

In problem (б) it is easier to obtain the solution under the condition (n1,n2)=0\left(\mathbf{n}_{1}, \mathbf{n}_{2}\right)=0.