Question

Find a vector equation and parametric equations for the line The line through the point (2, 2.4, 3.5) and parallel to the vector 3i 1 2j 2 k

Answer 1

Answer:

The vector equation

$r = (2 + 3t)i+ (2.4 + 2t)j+ (3.5 - t)k$

The parametric equation

$x = 2 + 3t\\y = 2.4 + 2t\\z = 3.5-t$

Step-by-step explanation:

Given

$Point = (2,2.4,3.5)$

$Vector = 3i + 2j - k$

Required

The vector equation

First, we calculate the position vector of the point.

This is represented as:

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

The vector equation is then calculated as:

$r = r_o + t * Vector$

$r = 2i + 2.4j + 3.5k + t * (3i + 2j - k)$

Open bracket

$r = 2i + 2.4j + 3.5k + 3ti + 2tj - tk$

Collect like terms

$r = 2i + 3ti+ 2.4j + 2tj+ 3.5k - tk$

Factorize

$r = (2 + 3t)i+ (2.4 + 2t)j+ (3.5 - t)k$

The parametric equation is represented as:

$x = x_0 + at\\y = y_0 + bt\\z = z_0 + ct$

Where

$r = (x_0 + at)i +(y_0 + bt)j+(z_0 + ct)k$

By comparison:

$x = 2 + 3t\\y = 2.4 + 2t\\z = 3.5-t$