Questions: Give a vector parametric equation for the line through the point P(0,1,3) that is parallel to the line <-4,2,1-3 t> : L(t)= (Enter your answer as <x0, y0, z0>+t<a, b, c> or <x0+a t, y0+b t, z0+c t>)

Give a vector parametric equation for the line through the point P(0,1,3) that is parallel to the line <-4,2,1-3 t> :
L(t)=
(Enter your answer as <x0, y0, z0>+t<a, b, c> or <x0+a t, y0+b t, z0+c t>)

Transcript text: 10:09 WeBWorK $M A A$ MATHEMATICAL ASSOCIATION OF AMERICA Logged in as Chloé Sabbah. Log Out WeBWorK / linalganna / 2024Set07 / 7 Previous Problem Problem List Next Problem 2024Set07: Problem 7 (1 point) Give a vector parametric equation for the line through the point $\mathrm{P}(0,1,3)$ that is parallel to the line $\langle-4,2,1-3 t\rangle$ : \[ L(t)= \] $\square$ (Enter your answer as $\left.\left\langle x_{0}, y_{0}, z_{0}\right\rangle+t\right)$ Preview My Answers Submit Answers You have attempted this problem 0 times. You have unlimited attempts remaining. Email Instructor AA gauss.vaniercollege.qc.ca

Solution

Solution Steps

Step 1: Identify the given points and direction vector

The problem provides two points:

Point $ P(0, 1, 3) $
Point $ Q(-4, 2, 1) $

To find the direction vector $\vec{d}$, subtract the coordinates of $P$ from $Q$: \[ \vec{d} = Q - P = (-4 - 0, 2 - 1, 1 - 3) = (-4, 1, -2) \]

Step 2: Write the parametric equation

The parametric equation of a line passing through point $P(x_0, y_0, z_0)$ with direction vector $\vec{d} = (a, b, c)$ is: \[ L(t) = \langle x_0, y_0, z_0 \rangle + t \langle a, b, c \rangle \]

Substitute $P(0, 1, 3)$ and $\vec{d} = (-4, 1, -2)$: \[ L(t) = \langle 0, 1, 3 \rangle + t \langle -4, 1, -2 \rangle \]