Question

find parametric equations for the line through the point ( p_0(-3, 2, 1) ) and parallel to the vector ( mathbf{v} = langle -1, -1, 2
angle ). use ( t ) as the parameter in your answers.
( x(t) = ) input box help (formulas)
( y(t) = ) input box help (formulas)
( z(t) = ) input box help (formulas)
submit answer next item

Explanation:

Step1: Recall parametric line formula

The parametric equations for a line in 3D space through point \( P_0(x_0, y_0, z_0) \) and parallel to vector \( \mathbf{v} = \langle a, b, c
angle \) are \( x(t)=x_0 + at \), \( y(t)=y_0 + bt \), \( z(t)=z_0 + ct \).

Step2: Identify \( x_0, y_0, z_0 \) and \( a, b, c \)

Given \( P_0(-3, 2, 1) \), so \( x_0 = -3 \), \( y_0 = 2 \), \( z_0 = 1 \). Given \( \mathbf{v} = \langle -1, -1, 2
angle \), so \( a = -1 \), \( b = -1 \), \( c = 2 \).

Step3: Find \( x(t) \)

Substitute \( x_0 = -3 \) and \( a = -1 \) into \( x(t)=x_0 + at \):
\( x(t) = -3 + (-1)t = -3 - t \)

Step4: Find \( y(t) \)

Substitute \( y_0 = 2 \) and \( b = -1 \) into \( y(t)=y_0 + bt \):
\( y(t) = 2 + (-1)t = 2 - t \)

Step5: Find \( z(t) \)

Substitute \( z_0 = 1 \) and \( c = 2 \) into \( z(t)=z_0 + ct \):
\( z(t) = 1 + 2t \)

Answer:

\( x(t) = -3 - t \)
\( y(t) = 2 - t \)
\( z(t) = 1 + 2t \)