The length of the projection of the line segment joining the points (5, 1, 4) and (4, 1, - Maths - Three Dimensional Geometry

Question

The length of the projection of the line segment joining the points
(5, –1, 4) and (4, –1, 3) on the x + y + z = 7
is
(A) 2 3 (B) 2 3 (C) 2 3 (D) 1 3

Ankita Agarwal · Accepted Answer

Dear Student,
<img src=
"https://s3mn%20mnimgs%20com/img/shared/content_ck_images/ck_5af16c9066cc7%20png">

Let points A5, -1, 4 and B4, -1,3AB=4-5i^+-1+1j^+3-4k^     =-i^^-k^AB=-12+-12          =2As given x+y+z=7so the normal plane n=i^+j^+k^n=12+12+12=3Equation of plane r
¯ n^=dxi^+yj^+zk^i^+j^+k^=7Now  the projection P=AB sinθ
iAs we know cosθ=AB
nABncos θ=-i^^-k^i^+j^+k^23         =-1-123         =-23          =-23As we know sin2θ+cos2θ=1sin θ=1-cos2θsinθ=1--232        =1-23         =13Now putting the value of sin θ and AB in equation iwe get Projection P=2
13                       =Hence Option (A) is correct 
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The length of the projection of the line segment joining the points (5, –1, 4) and (4, –1, 3) on the x + y + z = 7 is (A) 2 3 (B) 2 3 (C) 2 3 (D) 1 3

The length of the projection of the line segment joining the points (5, –1, 4) and (4, –1, 3) on the x + y + z = 7 is
(A) $\sqrt{\frac{2}{3}}$ (B) $\frac{2}{\sqrt{3}}$ (C) $\frac{2}{3}$ (D) $\frac{1}{3}$