First slide

Straight lines in 3D

Question

A line with positive direction cosines passes through the point $A (1, 2, - 1)$ and makes equal angles with the coordinate axes. The line meets the plane $x + 2 y + z = 5$ at the point $B$ . Then the point $B$ and length of the line segment $\bar{A B}$ respectively, are equal to

Moderate

A

$(\frac{- 3}{2}, \frac{5}{4}, 1); \frac{3}{4}$
B

$(- 5, 9, - 3), \sqrt{3}$
C

$(\frac{5}{4}, \frac{9}{4}, \frac{- 3}{4}); \frac{\sqrt{3}}{4}$
D

$(1, 2, - 1); 0$

Solution

Equation of line passing through the point   $A (1, 2, - 1)$ and making equal angles with axes, having positive direction ratios is $x - 1 = y - 2 = z + 1$
General point on the line is $(k + 1, k + 2, k - 1)$
Suppose that this point lies on the plane   $x + 2 y + z = 5$
It gives
             $\begin{matrix} k + 1 + 2 k + 4 + k - 1 = 5 \\ 4 k = 1 \\ k = \frac{1}{4} \end{matrix}$
Hence, the point of intersection of line and plane is $B (\frac{5}{4}, \frac{9}{4}, - \frac{3}{4})$
The distance between $A (1, 2, - 1), B (\frac{5}{4}, \frac{9}{4}, - \frac{3}{4})$ is $A B = \sqrt{\frac{1}{16} + \frac{1}{16} + \frac{1}{16}} = \frac{\sqrt{3}}{4}$

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