Question

find parametric equations for the line through the point ( p_0(0, 2, -4) ) and parallel to the line with parametric equations ( x = -4 - 2t ), ( y = -5 - 5t ), ( z = 1 - 4t ). use ( t ) as the parameter in your answers. ( x(t) = ) input box, ( y(t) = ) input box, ( z(t) = ) input box help (formulas) help (formulas) help (formulas) submit answer next item

Explanation:

Step1: Recall parametric line formula

The parametric equations of a line through point \( P_0(x_0, y_0, z_0) \) and parallel to the vector \( \vec{v} = \langle a, b, c
angle \) are \( x = x_0 + at \), \( y = y_0 + bt \), \( z = z_0 + ct \). Here, \( P_0(0, 2, -4) \) and the direction vector from the given line \( x = -4 - 2t \), \( y = -5 - 5t \), \( z = 1 - 4t \) is \( \vec{v} = \langle -2, -5, -4
angle \).

Step2: Find \( x(t) \)

Substitute \( x_0 = 0 \), \( a = -2 \) into \( x = x_0 + at \):
\( x(t) = 0 + (-2)t = -2t \)

Step3: Find \( y(t) \)

Substitute \( y_0 = 2 \), \( b = -5 \) into \( y = y_0 + bt \):
\( y(t) = 2 + (-5)t = 2 - 5t \)

Step4: Find \( z(t) \)

Substitute \( z_0 = -4 \), \( c = -4 \) into \( z = z_0 + ct \):
\( z(t) = -4 + (-4)t = -4 - 4t \)

Answer:

\( x(t) = -2t \), \( y(t) = 2 - 5t \), \( z(t) = -4 - 4t \)