Complete Question:
Find both the vector equation and the parametric equations of the line through (0,0,0) that is perpendicular to both [tex]u = < 2, 0, 2>[/tex] and [tex]w = < -2, 1, 0>[/tex] 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 [tex]u = < 2, 0, 2>[/tex] and [tex]w = < -2, 1, 0>[/tex] , we will find the cross product of u and w
[tex]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>[/tex]
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
The equation of the line and its parametric equations are [tex]\vec r = \langle 0,0,0 \rangle + t\cdot \langle -2,4,2 \rangle[/tex] and [tex](x,y, z) = (-2\cdot t, 4\cdot t, 2\cdot t)[/tex].
Vectorially speaking, the equation of the line is defined by the following formula:
[tex]\vec r = \vec O + t\cdot \vec l[/tex] (1)
Where:
Since [tex]\vec l[/tex] must be perpendicular both to [tex]\vec u[/tex] and [tex]\vec v[/tex], then we must apply cross product:
[tex]\vec l = \left|\begin{array}{ccc}\hat{i}&\hat{j}&\hat{k}\\2&0&2\\2&1&0\end{array}\right|[/tex]
[tex]\vec l = \langle -2, 4, 2\rangle[/tex] (2)
Then, we have the following equation of the line: ([tex]\vec O = \langle 0,0,0 \rangle[/tex], [tex]\vec l = \langle -2, 4, 2\rangle[/tex])
[tex]\vec r = \langle 0,0,0 \rangle + t\cdot \langle -2,4,2 \rangle[/tex] (3)
And the parametric equations are, respectively:
[tex](x,y, z) = (-2\cdot t, 4\cdot t, 2\cdot t)[/tex] (4)
The equation of the line and its parametric equations are [tex]\vec r = \langle 0,0,0 \rangle + t\cdot \langle -2,4,2 \rangle[/tex] and [tex](x,y, z) = (-2\cdot t, 4\cdot t, 2\cdot t)[/tex]. [tex]\blacksquare[/tex]
The statement presents typing mistakes and is poorly formatted. Correct form is shown below:
Find both vector equation and the parametric equations of the line through (0, 0, 0) that is perpendicular to the following two vectors: [tex]\vec u = \langle 2, 0, 2 \rangle[/tex], [tex]\vec v = (2, 1, 0)[/tex].
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