Question

Check whether the vector (2root3, 2 ) is equally inclined to the vectors (2, 2root3) and (4,0)

Answer 1

Answer on Question #81393 – Math – Linear Algebra

Question

Check whether the vector $(2\sqrt{3}, 2)$ is equally inclined to the vectors $(2, 2\sqrt{3})$ and $(4, 0)$.

Solution

Let $\vec{a} = (2\sqrt{3}, 2), \vec{b} = (2, 2\sqrt{3})$ and $\vec{c} = (4, 0)$.

Find dot product

$$\vec{a} \cdot \vec{b} = 2\sqrt{3}(2) + 2(2\sqrt{3}) = 8\sqrt{3}$$ $$\vec{a} \cdot \vec{c} = 2\sqrt{3}(4) + 2(0) = 8\sqrt{3}$$ $$|\vec{a}| = \sqrt{(2\sqrt{3})^2 + (2)^2} = 4$$ $$|\vec{b}| = \sqrt{(2)^2 + (2\sqrt{3})^2} = 4$$ $$|\vec{c}| = \sqrt{(4)^2 + (0)^2} = 4$$ $$\cos \beta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|} = \frac{8\sqrt{3}}{4 \cdot 4} = \frac{\sqrt{3}}{2}$$ $$\cos \gamma = \frac{\vec{a} \cdot \vec{c}}{|\vec{a}||\vec{c}|} = \frac{8\sqrt{3}}{4 \cdot 4} = \frac{\sqrt{3}}{2}$$

Therefore, the vector $(2\sqrt{3}, 2)$ is equally inclined to the vectors $(2, 2\sqrt{3})$ and $(4, 0)$.

Answer: the vector $(2\sqrt{3}, 2)$ is equally inclined to the vectors $(2, 2\sqrt{3})$ and $(4, 0)$.

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