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Two vectors a and b have equal magnitudes of 12 units. These vectors are making angles 30^o and 120^o with the x-axis respectively. Their sum is $\vec{r}$ . Find the x and y components of $\vec{r}$ .

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a

$(6 \sqrt{3} + 6), (6 - 6 \sqrt{3})$

b

$(6 \sqrt{3} - 6), (6 - 6 \sqrt{3})$

c

$(6 \sqrt{3} + 6), (6 + 6 \sqrt{3})$

d

$(6 \sqrt{3} - 6), (6 + 6 \sqrt{3})$

answer is A.

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Detailed Solution

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Here a = b =12 units
$X-component = acos 30^{\circ} - bcos 60^{\circ}$
$= 12 \times \frac{\sqrt{3}}{2} - 12 \times \frac{1}{2} \Rightarrow (6 \sqrt{3} - 6)$
$Y-component = asin 30^{\circ} + bsin 60^{\circ}$
$\begin{array}{l} = 12 \times \frac{1}{2} + 12 \times \frac{\sqrt{3}}{2} = (6 + 6 \sqrt{3}) \\ (x, y) = (6 \sqrt{3} - 6), (6 + 6 \sqrt{3}) \end{array}$

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