<p style="font-size: 1.125rem"><strong>SCALAR PRODUCT (DOT PRODUCT)</strong></p>
<div class="content-block content-block--table"><div class="table-container"><table><tbody><tr style="height: 23px;"><td rowspan="2" style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>${\mathbf{a} \cdot \mathbf{b} = \left| \mathbf{a} \right|\left| \mathbf{b} \right|\cos\theta}$</p></td></tr><tr style="height: 40px;"></tr></tbody></table></div></div>
<p>where ${\theta}$ is the angle between ${\mathbf{a}}$ and ${\mathbf{b}}$ when placed at the same initial points (or the same terminal points), ${0^{\circ} \leq \theta \leq 180^{\circ}}$.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1768615575882-lxtsnkiypl.png" style="max-width: min(170px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></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>Scalar Product in column vectors:</p><p>${\mathbf{a} \cdot \mathbf{b} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} \cdot \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} = a_1 b_1 + a_2 b_2 + a_3 b_3}$</p></div><div class="question-column" style="flex: 1; align-self: flex-start;"><p>For example,</p><p>${\begin{pmatrix} -2 \\ 1 \\ 4 \end{pmatrix} \cdot \begin{pmatrix} 2 \\ 5 \\ 0 \end{pmatrix} = -4 + 5 + 0 = 1}$</p></div></div></div>
<p style="font-size: 1.125rem"><strong>PROPERTIES OF SCALAR PRODUCT</strong></p>
<p>• ${\mathbf{a} \cdot \mathbf{b}}$ is a real number (scalar), not a vector.</p>
<p>• ${\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}}$ (Commutative law).</p>
<p>• If either ${\mathbf{a} = \mathbf{0}}$ or ${\mathbf{b} = \mathbf{0}}$, then ${\mathbf{a} \cdot \mathbf{b} = 0}$ (real number zero).</p>
<p>• ${\mathbf{a} \cdot \mathbf{a} = \left| \mathbf{a} \right|^2}$, since ${\mathbf{a}}$ is parallel to itself ${\Rightarrow \theta = 0^{\circ} \Rightarrow \cos 0^{\circ} = 1}$.</p>
<p>• ${\mathbf{a} \cdot \mathbf{b} &gt; 0}$ when ${0^{\circ} \leq \theta &lt; 90^{\circ}}$; ${\mathbf{a} \cdot \mathbf{b} &lt; 0}$ when ${90^{\circ} &lt; \theta \leq 180^{\circ}}$.</p>
<p>• For ${\lambda, \mu \in \mathbb{R}}$,</p>
<p style="margin-left: 2rem">1. ${\lambda(\mathbf{a} \cdot \mathbf{b}) = (\lambda \mathbf{a}) \cdot \mathbf{b} = \mathbf{a} \cdot (\lambda \mathbf{b})}$;</p>
<p style="margin-left: 2rem">2. ${\lambda \mathbf{a} \cdot \mu \mathbf{b} = (\lambda \mu) \mathbf{a} \cdot \mathbf{b}}$.</p>
<p>• Distributive law</p>
<p style="margin-left: 2rem">1. ${\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}}$.</p>
<p style="margin-left: 2rem">2. ${(\mathbf{a} + \mathbf{b}) \cdot \mathbf{c} = \mathbf{a} \cdot \mathbf{c} + \mathbf{b} \cdot \mathbf{c}}$.</p>
<p style="margin-left: 2rem">3. ${(\mathbf{a} + \mathbf{b}) \cdot (\mathbf{c} + \mathbf{d}) = \mathbf{a} \cdot \mathbf{c} + \mathbf{a} \cdot \mathbf{d} + \mathbf{b} \cdot \mathbf{c} + \mathbf{b} \cdot \mathbf{d}}$,<br></p>
<p style="margin-left: 5rem">specially, ${(\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} + \mathbf{b}) = \mathbf{a} \cdot \mathbf{a} + 2\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{b} = \left| \mathbf{a} \right|^2 + 2\mathbf{a} \cdot \mathbf{b} + \left| \mathbf{b} \right|^2}$.</p>
<div class="content-block content-block--table"><div class="table-container"><table><colgroup><col style="width: 440px;"></colgroup><tbody><tr style="height: 23px;"><td rowspan="2" style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"><p><strong>Common Mistake:</strong></p><p>The scalar product of vectors does NOT satisfy Associative law or Cancellation law:</p><p>${(\mathbf{a} \cdot \mathbf{b}) \cdot \mathbf{c} \neq \mathbf{a} \cdot (\mathbf{b} \cdot \mathbf{c})}$</p><p>${\mathbf{a} \cdot \mathbf{b} = \mathbf{a} \cdot \mathbf{c} \not\Rightarrow \mathbf{b} = \mathbf{c}}$</p></td></tr><tr style="height: 40px;"></tr></tbody></table></div></div>
<p style="font-size: 1.125rem"><strong>APPLICATIONS OF SCALAR PRODUCT</strong></p>
<p>PARALLELISM TEST</p>
<div class="content-block content-block--table"><div class="table-container"><table><tbody><tr style="height: 23px;"><td rowspan="2" style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"><p>${\mathbf{a} \cdot \mathbf{b} = \pm \left| \mathbf{a} \right| \left| \mathbf{b} \right| \iff \mathbf{a} \parallel \mathbf{b}}$</p></td></tr><tr style="height: 40px;"></tr></tbody></table></div></div>
<p>when ${\theta = 0^{\circ}}$ or ${180^{\circ}}$. Specifically,<br>when ${\theta = 0^{\circ} \Rightarrow \cos 0^{\circ} = 1}$, ${\mathbf{a}}$ and ${\mathbf{b}}$ are parallel and in the same direction;<br>when ${\theta = 180^{\circ} \Rightarrow \cos 180^{\circ} = -1}$, ${\mathbf{a}}$ and ${\mathbf{b}}$ are parallel and in the opposite direction.</p>
<p>PERPENDICULARITY TEST</p>
<div class="content-block content-block--table"><div class="table-container"><table><tbody><tr style="height: 23px;"><td rowspan="2" style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"><p>${\mathbf{a} \cdot \mathbf{b} = 0 \iff \mathbf{a} \perp \mathbf{b}}$</p></td></tr><tr style="height: 40px;"></tr></tbody></table></div></div>
<p>when ${\theta = 90^{\circ} \Rightarrow \cos 90^{\circ} = 0}$, ${\mathbf{a}}$ and ${\mathbf{b}}$ are perpendicular to each other.</p>
<p>ANGLE BETWEEN TWO VECTORS</p>
<div class="content-block content-block--table"><div class="table-container"><table><tbody><tr style="height: 23px;"><td rowspan="2" style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"><p>${\cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\left| \mathbf{a} \right| \left| \mathbf{b} \right|} \Rightarrow \theta = \cos^{-1} \left( \frac{\mathbf{a} \cdot \mathbf{b}}{\left| \mathbf{a} \right| \left| \mathbf{b} \right|} \right)}$</p></td></tr><tr style="height: 40px;"></tr></tbody></table></div></div>
<p>LENGTH OF PROJECTION</p>
<p>The length of projection of ${\mathbf{a}}$ onto ${\mathbf{b}}$, ${l}$, is given by ${l = \left| \mathbf{a} \right| \cos \theta = \left| \mathbf{a} \right| \frac{\mathbf{a} \cdot \mathbf{b}}{\left| \mathbf{a} \right| \left| \mathbf{b} \right|}}$, and so</p>
<div class="content-block content-block--table"><div class="table-container"><table><tbody><tr style="height: 23px;"><td rowspan="2" style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"><p>${l = \frac{\left| \mathbf{a} \cdot \mathbf{b} \right|}{\left| \mathbf{b} \right|} = \left| \mathbf{a} \cdot \hat{\mathbf{b}} \right|}$</p></td></tr><tr style="height: 40px;"></tr></tbody></table></div></div>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1768616956259-d555rqbtmlf.png" style="max-width: min(170px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

<p>${\mathbf{a} \cdot \mathbf{b} = \left| \mathbf{a} \right|\left| \mathbf{b} \right|\cos\theta}$</p>

where ${\theta}$ is the angle between ${\mathbf{a}}$ and ${\mathbf{b}}$ when placed at the same initial points (or the same terminal points), ${0^{\circ} \leq \theta \leq 180^{\circ}}$.

• ${\mathbf{a} \cdot \mathbf{b}}$ is a real number (scalar), not a vector.

• ${\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}}$ (Commutative law).

• If either ${\mathbf{a} = \mathbf{0}}$ or ${\mathbf{b} = \mathbf{0}}$, then ${\mathbf{a} \cdot \mathbf{b} = 0}$ (real number zero).

• ${\mathbf{a} \cdot \mathbf{a} = \left| \mathbf{a} \right|^2}$, since ${\mathbf{a}}$ is parallel to itself ${\Rightarrow \theta = 0^{\circ} \Rightarrow \cos 0^{\circ} = 1}$.

• ${\mathbf{a} \cdot \mathbf{b} > 0}$ when ${0^{\circ} \leq \theta < 90^{\circ}}$; ${\mathbf{a} \cdot \mathbf{b} < 0}$ when ${90^{\circ} < \theta \leq 180^{\circ}}$.

1. ${\lambda(\mathbf{a} \cdot \mathbf{b}) = (\lambda \mathbf{a}) \cdot \mathbf{b} = \mathbf{a} \cdot (\lambda \mathbf{b})}$;

2. ${\lambda \mathbf{a} \cdot \mu \mathbf{b} = (\lambda \mu) \mathbf{a} \cdot \mathbf{b}}$.

1. ${\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}}$.

2. ${(\mathbf{a} + \mathbf{b}) \cdot \mathbf{c} = \mathbf{a} \cdot \mathbf{c} + \mathbf{b} \cdot \mathbf{c}}$.

3. ${(\mathbf{a} + \mathbf{b}) \cdot (\mathbf{c} + \mathbf{d}) = \mathbf{a} \cdot \mathbf{c} + \mathbf{a} \cdot \mathbf{d} + \mathbf{b} \cdot \mathbf{c} + \mathbf{b} \cdot \mathbf{d}}$,


specially, ${(\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} + \mathbf{b}) = \mathbf{a} \cdot \mathbf{a} + 2\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{b} = \left| \mathbf{a} \right|^2 + 2\mathbf{a} \cdot \mathbf{b} + \left| \mathbf{b} \right|^2}$.

<p><strong>Common Mistake:</strong></p><p>The scalar product of vectors does NOT satisfy Associative law or Cancellation law:</p><p>${(\mathbf{a} \cdot \mathbf{b}) \cdot \mathbf{c} \neq \mathbf{a} \cdot (\mathbf{b} \cdot \mathbf{c})}$</p><p>${\mathbf{a} \cdot \mathbf{b} = \mathbf{a} \cdot \mathbf{c} \not\Rightarrow \mathbf{b} = \mathbf{c}}$</p>

<p>${\mathbf{a} \cdot \mathbf{b} = \pm \left| \mathbf{a} \right| \left| \mathbf{b} \right| \iff \mathbf{a} \parallel \mathbf{b}}$</p>

when ${\theta = 0^{\circ}}$ or ${180^{\circ}}$. Specifically,
when ${\theta = 0^{\circ} \Rightarrow \cos 0^{\circ} = 1}$, ${\mathbf{a}}$ and ${\mathbf{b}}$ are parallel and in the same direction;
when ${\theta = 180^{\circ} \Rightarrow \cos 180^{\circ} = -1}$, ${\mathbf{a}}$ and ${\mathbf{b}}$ are parallel and in the opposite direction.

<p>${\mathbf{a} \cdot \mathbf{b} = 0 \iff \mathbf{a} \perp \mathbf{b}}$</p>

when ${\theta = 90^{\circ} \Rightarrow \cos 90^{\circ} = 0}$, ${\mathbf{a}}$ and ${\mathbf{b}}$ are perpendicular to each other.

<p>${\cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\left| \mathbf{a} \right| \left| \mathbf{b} \right|} \Rightarrow \theta = \cos^{-1} \left( \frac{\mathbf{a} \cdot \mathbf{b}}{\left| \mathbf{a} \right| \left| \mathbf{b} \right|} \right)}$</p>

The length of projection of ${\mathbf{a}}$ onto ${\mathbf{b}}$, ${l}$, is given by ${l = \left| \mathbf{a} \right| \cos \theta = \left| \mathbf{a} \right| \frac{\mathbf{a} \cdot \mathbf{b}}{\left| \mathbf{a} \right| \left| \mathbf{b} \right|}}$, and so

<p>${l = \frac{\left| \mathbf{a} \cdot \mathbf{b} \right|}{\left| \mathbf{b} \right|} = \left| \mathbf{a} \cdot \hat{\mathbf{b}} \right|}$</p>

Summary Notes: Vectors

Summary Notes: Vectors

Video Solution: