Question

Find parametric equations and symmetric equations for the line. (Use the parameter t.) The line through (5, 4, 0) and perpendicular to both i j and j k

Answer 1

Explanation:

Suppose u = i + j =
`\Big \langle 1,1,0 \Big\rangle` & v = j + k =
`\Big \langle 0,1,1 \Big\rangle`

The direction vector for the line u*v is:

`u* v = \left |\begin{array}{ccc}i&j&k\\1&1&0\\0&1&1\end {array} \right|`

= (1-0) i - (1 - 0)j + ( 1- 0) k

= i - j - k

=
`\Big \langle 1,-1,1 \Big\rangle`

Hence, the equation of the line via the point (5,4,0 ) and the direction vector
`\Big \langle 1,-1,1 \Big\rangle` is as follows:

r(t) = (5,4,0) + t
`\Big \langle 1,-1,1 \Big\rangle`

r(t) = (5+t, 4-t, t)

The symmetric equations of the line are:

`(x-5)/(1)= (y-4)/(-1) = (z)/(1)`

x - 5 = -(y-4) = z

The parametric equation of the line is:

`(x-5)/(1)= (y-4)/(-1) = (z)/(1)= t`

x = 5 + t , y = 4 - t , z = t