Addition Of Vectors

Step 1: Using the cosine rule. We are given the sides of the triangle as: To find the greatest angle, we use the cosine rule: Substituting the values of , , and , we get: Using the identity , this simplifies to: Therefore, .

&\text{Given, } |\vec{a}| = \sqrt{3}, |\vec{b}| = 5, |\vec{c}| = 10
&\text{Using the vector triple product: } \vec{b} \text{ and } \vec{c} = \frac{\pi}{3}
&\vec{b} \cdot \vec{c} = |\vec{b}||\vec{c}| \cos \left(\frac{\pi}{3}\right) = 5 \times |\vec{c}| \times \frac{1}{2} = 10
&|\vec{c}| = 4
&\text{and } \vec{a} \text{ is perpendicular to } \vec{b} \times \vec{c}
&\text{now } \vec{a} \cdot (\vec{b} \times \vec{c}) = 0
&|\vec{a} \times (\vec{b} \times \vec{c})| = |\vec{a}||\vec{b} \times \vec{c}| \sin \frac{\pi}{2}
&= \sqrt{3} \times |\vec{b}||\vec{c}| \sin \frac{\pi}{3} \sin \frac{\pi}{2}
&= \sqrt{3} \times 5 \times 4 \times \frac{\sqrt{3}}{2}
&= 30