If the vectors x-k-y-k- and -zk- are co-planar where x- 1-y- 1-z- 1- then prove that 11-x-11-y-11-z-1-

Since, #160;Three given vectors are coplanar, #160;Then, #160; [ | [ x amp; 1 amp; 1; 1 amp; y amp; 1; 1 amp; 1 amp; z ]| =0; ...

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Equation of a Plane : Normal Form

If the vector...

Question

If the vectors $x^i +^j +^k,^i + y^j +^k$ and $^i +^j + z^k$ are co-planar where $x \neq 1, y \neq 1, z \neq 1$ , then prove that $\frac{1}{1 - x} + \frac{1}{1 - y} + \frac{1}{1 - z} = 1$ .

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^i \times ((^a -^j) \times^i) +^j \times ((^a -^k) \times^j) +^k \times ((^a -^i) \times^k) = \to 0

\to a = x^i + y^j + z^k,

8 (x^{3} - x y + z x)

a^i +^j +^k,^i + b^j +^k,^i +^j + c^k

(a \neq b \neq c \neq 1)

\frac{1}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c} =

a^i +^j +^k,^i + b^j +^k,^i +^j + c^k (a \neq 1, b \neq 1, c \neq 1)

\frac{1}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c}

α^i +^j +^k,^i + β^j +^k,^i +^j + λ^k (α, β, γ \neq 1)

\frac{1}{1 - α} + \frac{1}{1 + β} + \frac{1}{1 - γ}

\to p = a^i +^j +^k, \to q =^i + b^j +^k, \to r =^i +^j + c^k

a, b, c \neq 1

\frac{1}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c} = 1

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