if-a-b-c-d-R-verify-a-2b-3c-4d-6-then-find-min-a-2-b-2-c-2-d-2-

Question Number 155495 by mathdanisur last updated on 01/Oct/21

if a; b; c; d \in R verify a + 2 b + 3 c + 4 d = 6

then find \min (a^{2} + b^{2} + c^{2} + d^{2})

Answered by mr W last updated on 01/Oct/21

{(a^{2} + b^{2} + c^{2} + d^{2})}_{m i n} = {(\frac{6}{\sqrt{1^{2} + 2^{2} + 3^{2} + 4^{2}}})}^{2} = \frac{6}{5}

Commented by mathdanisur last updated on 01/Oct/21

Thanlyou S er, can you tell me how this

part was obtained, if possible

Commented by mr W last updated on 01/Oct/21

\sqrt{x^{2} + y^{2} + z^{2} + w^{2}} i s t h e d i s t a n c e f r o m

p o i n t (x, y, z, w) t o t h e o r i g i n (0, 0, 0, 0)

i n t h e 4 D s p a c e .

x + 2 y + 3 z + 4 w = 6 i s a n p l a n e i n t h e

4 D s p a c e .

t h e s h o r t e s t d i s t a n c e f r o m a p o i n t

(x, y, z, w) o n t h i s p l a n e t o t h e o r i g i n i s

{(\sqrt{x^{2} + y^{2} + z^{2} + w^{2}})}_{m i n} = \frac{∣ 0 + 2 \times 0 + 3 \times 0 + 4 \times 0 - 6 ∣}{\sqrt{1^{2} + 2^{2} + 3^{2} + 4^{4}}} = \frac{6}{\sqrt{30}}

t h e r e f o r e

{(a^{2} + b^{2} + c^{2} + d^{2})}_{m i n} = {(\sqrt{a^{2} + b^{2} + c^{2} + d^{2}})}_{m i n}^{2}

= {(\frac{6}{\sqrt{30}})}^{2} = \frac{6}{5}

Commented by mathdanisur last updated on 01/Oct/21

Creativ solution thank you S er