(a) Compute ‖u‖, ‖v‖, and u ∙ v for the given vector in R3. (b) Verify the Cauchy-Schwarz inequality for the given pair of vector.

Question

$= \sqrt{3 i} + π + c k, v = 4 i - j - k$ , where is a constant.
(a) Compute $‖ u ‖, ‖ v ‖$ , and $u \times v$ for the given vector in R3.
(b) Verify the Cauchy-Schwarz inequality for the given pair of vector.

Answer 1

Here $u = \sqrt{3} i + π j + c k, v = 4 i - j - k,$ where c is a constant. (a) $| | u | | = \sqrt{{\sqrt{3}}^{2} + π^{2} + c^{2}} = \sqrt{3 + π^{2} + c^{2}}$
$| | v | | = \sqrt{4^{2} + 1 + 1} = \sqrt{18}$
and
$u \times v = (\sqrt{3 i} + π j + c k) \times (4 i - j - k) = 4 \sqrt{3} - π - c .$
(b) $| u \times v | = | 4 \sqrt{3} - π - c | \leq \sqrt{3 + π^{2} + c^{2}} \sqrt{1} 8 = | | u | | | | v | | .$

(a) Compute ‖u‖, ‖v‖, and u ∙ v for the given vector in R3. (b) Verify the Cauchy-Schwarz inequality for the given pair of vector.

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