Question

Please answer ASAP with explanation.

Vectors $\vec{A}$ and $\vec{B}$ each have magnitude L. When drawn with their tails at the same point, the angle between them is 30°. The value of $\vec{A}$ • $\vec{B}$ is :

A. Zero
B. L²
C. $\sqrt{3} {L}^{2} /2$
D. 2L²​

Answer 1

$\maltese \: \large \underbrace{ \pmb { \frak \orange{Given \: that : - }}}$

• Vectors $\vec{A}$ and $\vec{B}$ each have magnitude L. When drawn with their tails at the same point, the angle between them is 30°

$\maltese \: \large \underbrace{ \pmb { \frak \orange{ Asked \: to \: calculate : - }}}$

• The value of $\vec{A}$ • $\vec{B}$ is ?

$\maltese \: \large \underbrace{ \pmb { \frak \orange{ Concept \: to \: be \: applied : - }}}$

• When two vectors (A and B) have a particular angle $(\theta)$ between them, then their Dot product is as follows :-

$\leadsto \underline{\boxed{\sf{ \pink{{ \overrightarrow{A}. \overrightarrow{B} = AB \cos \theta}}}}} \bigstar$

• Here, We are already provided with the magnitudes of both the vectors which is same and is equal to (L)

• $\sf\theta$ is also given as 30⁰

$\\ \dag\underline{ \frak{substituting \: the \: given \: values \: in \: the \: formula : - }}$

$\\ : \implies \sf \overrightarrow{A} .\overrightarrow{B} = (L)(L) \cos {30}^{0}$

$\\ : \implies \sf \overrightarrow{A} .\overrightarrow{B} = {L}^{2} \times \frac{ \sqrt{3} }{2}$

$\\ : \implies \boxed{\sf \pink{ \overrightarrow{A} .\overrightarrow{B} = \frac{ \sqrt{3} .{L}^{2} }{2} } }\bigstar$

$\therefore\underline {\sf (c) \: option \: is \: correct \: }$

______________________

hope it helps !

Answer 2

$\rule{200}5$

Need to FinD :-

• The dot product of the vectors .

$\red{\frak{Given}}\begin{cases} \sf \vec{A} \ and \ \vec{B} \ each \ have \ magnitude\ L .\\\\\sf The \ angle \ between\ them \ is \ 30^o .\end{cases}$

We know that the dot product of two vectors say A and B is given by ,

$\sf\longrightarrow \purple{\small \vec{A} .\vec{B}= |A| |B| \ cos\theta}$

• where θ is the angle between the two vectors .
• |A| and |B| are the magnitude of A and B respectively .

Here ,

• |A| = L
• |B| = L
• θ = 30°

Substituting the respective values ,

$\sf\longrightarrow \footnotesize \vec{A} .\vec{B}= |A| |B| \ cos\theta \\\\\\\sf\longrightarrow \footnotesize \vec{A} .\vec{B}= L \times L \times cos 30^o \\\\\\\sf\longrightarrow \footnotesize \vec{A} .\vec{B}= L^2\times \dfrac{\sqrt3}{2}\qquad\Bigg\lgroup \bf cos30^o =\dfrac{\sqrt3}{2}\Bigg\rgroup \\\\\\\sf\longrightarrow \footnotesize \underline{\underline{\red{ \vec{A} .\vec{B}= \dfrac{ \sqrt3 L^2}{2} }}}$

$\rule{200}5$