Complete Question:
Find both the vector equation and the parametric equations of the line through (0,0,0) that is perpendicular to both
and
where t = 0 corresponds to the first given point.
Answer:
Vector equation: (x, y, z) = (0, 0, 0) + t ( u x w)
Parametric equation:
x = -2t
y = -4t
z = 2t
Step-by-step explanation:
Since the line is perpendicular to
and
, we will find the cross product of u and w
![u \times w = \left[\begin{array}{ccc}i&j&k\\2&0&2\\-2&1&0\end{array}\right] \\\\u \times w = i(0-2) -j(0+4) + k(2)\\\\u \times w = -2i - 4j + 2k\\\\u \times w = < -2, -4, 2>](https://tex.z-dn.net/?f=u%20%5Ctimes%20w%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Di%26j%26k%5C%5C2%260%262%5C%5C-2%261%260%5Cend%7Barray%7D%5Cright%5D%20%5C%5C%5C%5Cu%20%5Ctimes%20w%20%3D%20i%280-2%29%20-j%280%2B4%29%20%2B%20k%282%29%5C%5C%5C%5Cu%20%5Ctimes%20w%20%3D%20-2i%20-%204j%20%2B%202k%5C%5C%5C%5Cu%20%5Ctimes%20w%20%3D%20%3C%20-2%2C%20-4%2C%202%3E)
The equation of the line can be given by:
(x, y, z) = (0, 0, 0) + t ( u x w)
(x, y, z) = (0, 0, 0) + t < -2, -4, 2 >
x = -2t, y = -4t, z = 2t