3D vectors: lines and planes

Vector playground

<div class="content-box-placement content-box-placement--full content-box-placement--align-left" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-left-width:3px"><div class="content-box-title">Definition and orientation</div><p>\[\mathbf a\times\mathbf b=|\mathbf a||\mathbf b|\sin\theta\,\widehat{\mathbf n},\qquad \widehat{\mathbf n}\perp\mathbf a,\mathbf b\]</p></div></div>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><p class="paragraph-header">Right-hand orientation</p><div class="graph-block" data-graph-config="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" data-width="300" data-height="200" style="width:300px;height:200px;max-width:100%;display:block;margin:0 auto;"></div></div><div class="question-column" style="flex: 1; align-self: flex-start;"><p></p></div></div></div>
<div class="content-box-placement content-box-placement--full content-box-placement--align-left" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-left-width:3px"><div class="content-box-title">Component determinant</div><p>\[\mathbf a\times\mathbf b=\begin{vmatrix}\mathbf i&\mathbf j&\mathbf k\\a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix},\qquad \mathbf b\times\mathbf a=-(\mathbf a\times\mathbf b).\]</p></div></div>
<p>For nonzero vectors, \(\mathbf a\times\mathbf b=\mathbf0\) exactly when they are parallel. The product is distributive but not commutative or associative; cancellation is not generally valid.</p>

\[\mathbf a\times\mathbf b=|\mathbf a||\mathbf b|\sin\theta\,\widehat{\mathbf n},\qquad \widehat{\mathbf n}\perp\mathbf a,\mathbf b\]

Definition and orientation

\[\mathbf a\times\mathbf b=\begin{vmatrix}\mathbf i&\mathbf j&\mathbf k\\a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix},\qquad \mathbf b\times\mathbf a=-(\mathbf a\times\mathbf b).\]

Component determinant

For nonzero vectors, \(\mathbf a\times\mathbf b=\mathbf0\) exactly when they are parallel. The product is distributive but not commutative or associative; cancellation is not generally valid.

Vector Product

Vector Product

Video Solution:

Solution:

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