let u^→ =i^→ −j^→ +k^→ and v^→ =2i^→ +j^→ +3k^→ (o,i,j,k) orthonormal 1) calculate ∣∣u^→ ∣∣ ,∣∣v^→ ∣∣ ,u^→ .v^→ 2) calculate cos(u^→ ,v^→ ) 3)calculate u^→ Λv^→ and sin(u^→ ,v^→ )


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Question Number 86406 by mathmax by abdo last updated on 28/Mar/20

$l e t \vec{u} = \vec{i} - \vec{j} + \vec{k} a n d \vec{v} = 2 \vec{i} + \vec{j} + 3 \vec{k}$ $(o, i, j, k) o r t h o n o r m a l$ $1) c a l c u l a t e ∣∣ \vec{u} ∣∣, ∣∣ \vec{v} ∣∣, \vec{u} . \vec{v}$ $2) c a l c u l a t e c o s (\vec{u}, \vec{v})$ $3) c a l c u l a t e \vec{u} Λ \vec{v} a n d s i n (\vec{u}, \vec{v})$
	Answered by jagoll last updated on 28/Mar/20

	$∣∣ \vec{u} ∣∣ = \sqrt{3}$ $∣∣ \vec{v} ∣∣ = \sqrt{14}$ $\vec{u} . \vec{v} = 2 - 1 + 3 = 4$ $\cos (\vec{u}, \vec{v}) = \frac{4}{\sqrt{42}}$
	Answered by Rio Michael last updated on 28/Mar/20

	$3) ∣ \vec{u} \times \vec{v} ∣ = ∣ \vec{u} ∣ ∣ \vec{v} ∣ \sin θ$ $\Rightarrow \sin θ = \frac{∣ \vec{u} \times \vec{v} ∣}{∣ \vec{u} ∣∣ \vec{v} ∣}$ $\vec{u} \times \vec{v} = \| \begin{matrix} \vec{i} & \vec{j} & \vec{k} \\ 1 & - 1 & 1 \\ 2 & 1 & 3 \end{matrix} \| = i \| \begin{matrix} - 1 & 1 \\ 1 & 3 \end{matrix} \| - j \| \begin{matrix} 1 & 1 \\ 2 & 3 \end{matrix} \| + k \| \begin{matrix} 1 & - 1 \\ 2 & 1 \end{matrix} \|$ $= - 4 i - j + 3 k$ $∣ \vec{u} \times \vec{v} ∣ = \sqrt{26}$ $∣ \vec{u} ∣ = \sqrt{3} and ∣ \vec{v} ∣ = \sqrt{14} \Rightarrow \sin (\vec{u}, \vec{v}) = \frac{\sqrt{26}}{\sqrt{3} \sqrt{14}} = \sqrt{\frac{13}{14}}$
	Commented by abdomathmax last updated on 28/Mar/20

	$t h a n k x s i r .$
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