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Question

Three lines are given by

r=λi^,λR

r=μi^+j^,μR and

r=vi^+j^+k^,vR

Let the lines cut the plane x+y+z=1 at the points A,B and C respectively.

If the area of the triangle ABC is then the value of (6)2 equals __________.


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Solution

Step 1. Finding point A:

r=λi^ and x+y+z=1, we get

λ+0+0=1

λ=1

A=1,0,0

Step 2. Finding point B:

r=μi^+j^ and x+y+z=1, we get

μ+μ+0=1

μ=12

B=12,12,0

Step 3. Finding point C:

r=vi^+j^+k^ and x+y+z=1, we get

v+v+v=1

v=13

C=13,13,13

Step 4. Area of ABC=12|AB×AC|

Now, AB=-12i^+12j^and AC=-23i^+13j^+13k^

=12-12i^+12j^×-23i^+13j^+13k^=312

(6)2=63122=34=0.75

Hence, the value of (6)2 is 0.75


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