If vectors vec{a} and vec{b} represent the si | vedclass

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(C) In a regular hexagon $ABCDEF$,the sides are equal in magnitude and parallel to opposite sides.Given $\vec{AB} = \vec{a}$ and $\vec{BC} = \vec{b}$.

Vedclass · Accepted Answer

$\vec{a} - \vec{b}$

Vedclass · Answer

(C) In a regular hexagon $ABCDEF$,the sides are equal in magnitude and parallel to opposite sides.Given $\vec{AB} = \vec{a}$ and $\vec{BC} = \vec{b}$.By the triangle law of vector addition in $	riangle ABC$,$\vec{AC} = \vec{AB} + \vec{BC} = \vec{a} + \vec{b}$.In a regular hexagon,the vector $\vec{AD}$ is parallel to $\vec{BC}$ and has twice the magnitude,so $\vec{AD} = 2\vec{BC} = 2\vec{b}$.Also,$\vec{CD} = \vec{AD} - \vec{AC} = 2\vec{b} - (\vec{a} + \vec{b}) = \vec{b} - \vec{a}$.Since $\vec{FA}$ is parallel and equal in magnitude to $\vec{CD}$ in the opposite direction,or by observing the geometry,$\vec{FA} = \vec{CD} = \vec{b} - \vec{a}$ is incorrect based on the orientation; let us re-evaluate.Actually,$\vec{FA} = \vec{ED} = \vec{BC} - \vec{AB} = \vec{b} - \vec{a}$ is not correct. Let's use the property $\vec{AB} + \vec{BC} + \vec{CD} + \vec{DE} + \vec{EF} + \vec{FA} = 0$.Since $\vec{CD} = \vec{b} - \vec{a}$,$\vec{DE} = -\vec{a}$,$\vec{EF} = -\vec{b}$,$\vec{FA} = \vec{a} - \vec{b}$.

Column-$I$	Column-$II$
$(A)$ In $R^2$,if the magnitude of the projection vector of the vector $\alpha \hat{i}+\beta \hat{j}$ on $\sqrt{3} \hat{i}+\hat{j}$ is $\sqrt{3}$ and if $\alpha=2+\sqrt{3} \beta$,then possible value$(s)$ of $\|\alpha\|$ is (are)	$(P)$ $1$
$(B)$ Let $a$ and $b$ be real numbers such that the function $f(x)=\begin{cases} -3ax^2-2, & x < 1 \\ bx+a^2, & x \geq 1 \end{cases}$ is differentiable for all $x \in R$. Then possible value$(s)$ of $a$ is (are)	$(Q)$ $2$
$(C)$ Let $\omega \neq 1$ be a complex cube root of unity. If $(3-3\omega+2\omega^2)^{4n+3} + (2+3\omega-3\omega^2)^{4n+3} + (-3+2\omega+3\omega^2)^{4n+3}=0$,then possible value$(s)$ of $n$ is (are)	$(R)$ $3$
$(D)$ Let the harmonic mean of two positive real numbers $a$ and $b$ be $4$. If $q$ is a positive real number such that $a, 5, q, b$ is an arithmetic progression,then the value$(s)$ of $\|q-a\|$ is (are)	$(S)$ $4$
	$(T)$ $5$

If vectors $\vec{a}$ and $\vec{b}$ represent the sides $\vec{AB}$ and $\vec{BC}$ of a regular hexagon $ABCDEF$,find the vector represented by $\vec{FA}$.

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