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Expert-verified Found in: Page 572 ### Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280 # Find a vector equation for a line through the point $$(2,2.4,3.5)$$ and parallel to the vector $$3i + 2j - k$$ and the parametric equations for a line through the point $$(2,2.4,3.5)$$ and parallel to the vector $$3i + 2j - k$$.

The vector equation for a line through the point $$(2,2.4,3.5)$$ and parallel to the vector $$3i + 2j - k$$ is $$(2 + 3t)i + (2.4 + 2t)j + (3.5 - t)k$$

The parametric equations for a line through the point $$(2,2.4,3.5)$$ and parallel to the vector $$3i + 2j - k$$ are $$x = 2 + 3t,y = 2.4 + 2t,z = 3.5 - t$$

See the step by step solution

## Step 1: Write the expression to find vector equation $$(r)$$.

$$r = {r_0} + tv(1)$$

Here,

$${r_0}$$ is the position vector of point $${P_0}(2,2.4,3.5)$$ and

$$v$$ is the line's parallel vector.

## Step 2: Write the expressions to find the parametric equations for a line through the point $$\left( {{x_0},{y_0},{z_0}} \right)$$ and parallel to the direction vector $$ai + bj - ck$$.

$$\begin{array}{l}x = {x_0} + at,\\y = {y_0} + bt,\\z = {z_0} + ct{\rm{ (2) }}\end{array}$$

The position vector $$\left( {{r_0}} \right)$$ using the point $${P_0}(2,2.4,3.5)$$.

$${r_0} = 2i + 2.4j + 3.5k$$

## Step 3: Calculate the vector equation $$(r)$$.

In equation (1), substitute $$2i + 2.4j + 3.5k$$ for $${r_0}$$ and $$3i + 2j - k$$ for $$v$$.

$$\begin{array}{l}r = (2i + 2.4j + 3.5k) + t(3i + 2j - k)\\r = (2 + 3t)i + (2.4 + 2t)j + (3.5 - t)k\end{array}$$

Thus, the vector equation for a line through the point $$(2,2.4,3.5)$$ and parallel to the vector $$3i + 2j - k$$ is $$(2 + 3t)i + (2.4 + 2t)j + (3.5 - t)k$$

In equation (2), substitute 2 for $${x_0},2.4$$ for $${y_0},3.5$$ for $${z_0},3$$ for a, 2 for $$b$$, and $$- 1$$ for $$c$$.

$$\begin{array}{l}x = 2 + 3t,\\y = 2.4 + 2t,\\z = 3.5 + ( - 1)t\\x = 2 + 3t,\\y = 2.4 + 2t,\\z = 3.5 - t\end{array}$$

Thus, the parametric equations for a line through the point $$(2,2.4,3.5)$$ and parallel to the vector $$3i + 2j - k$$are

$$\begin{array}{l}x = 2 + 3t,\\y = 2.4 + 2t,\\z = 3.5 - t\end{array}$$ ### Want to see more solutions like these? 