Let P be a point not on the line L that passes through the points Q and R. The distance d from the point P to the line L is d =

Question

13 days ago

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Let P be a point not on the line L that passes through the points Q and R. The distance d from the point P to the line L is d =

|a × b| |a| where a = QR and b = QP. Use the above formula to find the distance from the point to the given line. (0, 1, 1); x = 2t, y = 5 − 2t, z = 1 + t

Mathematics

1 answer:

AnnZ [9K] · Answer 1 · 2026-03-14T15:42:55+03:00

Answer:

The distance from the point (0,1,1) to the specified line is zero.

Step-by-step explanation:

Considering the parametric equations of the line,

x=2t, y=5-2t, z=1+t

In order to calculate the distance from (0,1,1), we must remove t from the equations above, such that

$y=5-2t=5-x\implies x+y=5=4+1=4+z-t=4+z-\frac{1}{2}x$

$\implies 3x+2y-2z-8=0\hfill (1)$

whose direction ratios are (l,m,n)=(3,2,-2) and the distance from point (a,b,c)=(0,1,1) is defined as

$\frac{al+bm+cn}{\sqrt{l62+m^2+n^2}}=\frac{(3\times 0)+(2\times 1)+(-2)(1)}{\sqrt{3^2+2^2+(-2)^2}}=\frac{0+2-2}{\sqrt{17}}=0$

The distance between the point (0,1,1) and (1) amounts to zero. Therefore, the point (0,1,1) is located on the line (1).