Question

answer for this question ​

Answer 1

Answer:

: If the vectors ai + j + k, i + bj + k and i + j + ck (a ≠ b ≠ c ≠ 1) are coplanar, then the value of [1] / [1 − a] + [1] / [1 − b] + [1] / [1 − c] = _________.

Solution:

Since \begin{vmatrix} a &1 &1 \\ 1&b &1 \\ 1&1 &c\end{vmatrix}

∣

∣

∣

∣

∣

∣

∣

a

1

1

1

b

1

1

1

c

∣

∣

∣

∣

∣

∣

∣

= 0

Applying R2 → R2 − R1 and R3 → R3 − R1, we get

\begin{vmatrix} a &1 &1 \\ 1-a&b-1 &0 \\ 1-a&0 &c-1\end{vmatrix}

∣

∣

∣

∣

∣

∣

∣

a

1−a

1−a

1

b−1

0

1

0

c−1

∣

∣

∣

∣

∣

∣

∣

= 0

On expanding, we get

a (b − 1) (c − 1) − (1 − a) (c − 1) − (1 − a) (b − 1) = 0

On dividing by (1 − a) (1 − b) (1 − c), we get

[a] / [1 − a] + [1] / [1 − b] + [1] / [1 − c] = 0

⇒ [1] / [1 − a] + [1] / [1 − b] + [1] / [1 − c]

= {[1] / [1 − a]} − {[a] / [1 − a]}

= 1

Explanation:

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Answer 2

r=√2²+3²+5²

r=√4+9+25

r=√38