Given that u-2- -3k- v-2- -4k- -w-3- -3k- and u-R-15-v-R-30-w-R-20k-0- Then the greatest integer less than or equal to -R - is-

Given :u=î 2ĵ +3k̂, v=2î +ĵ+4k̂ , w=î+3ĵ +3k̂ and (u.R 15)î+(v.R 30)ĵ+(w.R 20)k̂=0 Let R=xî+yĵ+zk̂ So, u.R=15⇒ x 2y+3z=15⋯ (i) v.R=30⇒ 2x+y+4z=30⋯ (ii) w.R=25⇒ ...

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Byju's Answer

Standard XII

Mathematics

Intersection of Sets

Given that u=...

Question

Given that $\to u =^i - 2^j + 3^k, \to v = 2^i +^j + 4^k, \to w =^i + 3^j + 3^k$ and $(\to u . \to R - 15)^i + (\to v . \to R - 30)^j + (\to w . \to R - 20)^k = \to 0$ . Then the greatest integer less than or equal to $| \to R |$ is:

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Solution

Given : $\to u =^i - 2^j + 3^k, \to v = 2^i +^j + 4^k, \to w =^i + 3^j + 3^k$ and $(\to u . \to R - 15)^i + (\to v . \to R - 30)^j + (\to w . \to R - 20)^k = \to 0$
Let $\to R = x^i + y^j + z^k$
So, $\to u . \to R = 15 \Rightarrow x - 2 y + 3 z = 15 \dots (i)$
$\to v . \to R = 30 \Rightarrow 2 x + y + 4 z = 30 \dots (i i)$
$\to w . \to R = 25 \Rightarrow x + 3 y + 3 z = 25 \dots (i i i)$
On solving, we get: $x = 4, y = 2$ and $z = 5$
Hence, $| \to R | = \sqrt{16 + 4 + 25} = \sqrt{45}$
So, the greatest integer less than or equal to $| \to R |$ is $6$

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Given that →u=^i−2^j+3^k,→v=2^i+^j+4^k,→w=^i+3^j+3^k and (→u.→R−15)^i+(→v.→R−30)^j+(→w.→R−20)^k=→0. Then the greatest integer less than or equal to |→R| is:

Given that $\to u =^i - 2^j + 3^k, \to v = 2^i +^j + 4^k, \to w =^i + 3^j + 3^k$ and $(\to u . \to R - 15)^i + (\to v . \to R - 30)^j + (\to w . \to R - 20)^k = \to 0$ . Then the greatest integer less than or equal to $| \to R |$ is: