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Examples for Lines in space

Example 1

Find a parametric vector equation and Cartesian equations of the line through (1, 3,-2) parallel to 5i - 7j + 3k.

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Solution

According to our assumptions r0 = i + 3j - 2k is a position vector of a point on the line. Hence the parametric vector equation of the line is
r- r0 = (xi + yj + zk) - (i + 3j - 2k) = t(5i- 7j + 3k)

or equivalently

(x - 1)i + (y - 3)j + (z + 2)k = t(5i - 7j + 3k).

The parametric equations are then

(x- 1) = 5t, y - 3 = - 7t, z + 2 = 3t

so the Cartesian equations are

x- 1 y - 3 z + 2 ------= ------= -----. 5 - 7 3

Example 2

Find the parametric vector equation, and the cartesian equations, of the line through A(7, 4, 2) and B(8, 6, 5).

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Solution

The required line passes through the point A(7, 4, 2) and is parallel to the vector
-AB--> = (8- 7)i + (6 - 4)j + (5 - 2)k = i + 2j + 3k.

Its parametric vector equation is therefore

---> r- AB = (xi + yj + zk) - (7i + 4j + 2k) = t(i + 2j + 3k)

or

(x - 7)i + (y - 4)j + (z - 2)k = ti + 2tj + 3tk.

So the parametric equations of the line are

x - 7 = t, y- 4 = 2t, z - 2 = 3t

and the Cartesian equations are

x---7-= y---4-= z---2. 1 2 3