Question

The position vectors of the points A and B are (1,5,3) and (3,6,6). Find the vector equation of the line AB and the points where the line intersects the coordinate planes.

Answer 1

Answer on Question #52770, Math, Other

Task: The position vectors of the points A and B are (1,5,3) and (3,6,6). Find the vector equation of the line AB and the points where the line intersects the coordinate planes.

Answer:

Find a direction vector.

$$\vec {m} = O \vec {B} - O \vec {A} = [ 3, 6, 6 ] - [ 1, 5, 3 ] = [ 2, 1, 3 ]$$

the vector equation of the line AB is $[x,y,z] = [1,5,3] + t\cdot [2,1,3]$

so $x = 1 + 2t$ ; $y = 5 + t$ ; $z = 3 + 3t$ .

for the x-z plane, $y = 0$ , so $5 + t = 0$ , meaning that $t = -5$ . So We get $(-9,0,-12)$ .

for the x-y plane, $z = 0$ , so $3 + 3t = 0$ , meaning that $t = -1$ . So We get $(-1,4,0)$ .

for the y-z plane, $x = 0$ , so $1 + 2t = 0$ , meaning that $t = -1/2$ . So We get $(0,9/2,3/2)$ .

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