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Q3E

Expert-verifiedFound in: Page 572

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 \(x = 2 + 3t,y = 2.4 + 2t,z = 3.5 - t\)

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

Here,

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

\(v\) is the line's parallel vector.

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

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

\({r_0} = 2i + 2.4j + 3.5k\)

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

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

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