Find both the vector equation and the parametric equations of the line through (0,0,0) that is perpendicular to both $u = < 2, 0, 2>$ and $w = < -2, 1, 0>$ 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 $u = < 2, 0, 2>$ and $w = < -2, 1, 0>$ , we will find the cross product of u and w

$u \times w = \left[\begin{array}{ccc}i&j&k\\2&0&2\\-2&1&0\end{array}\right] \\\\u \times w = i(0-2) -j(0+4) + k(2)\\\\u \times w = -2i - 4j + 2k\\\\u \times w = < -2, -4, 2>$

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

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