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Question

Write a unit vector in the direction of the sum of the vector $$\vec {a} = 2\hat {i} + 2\hat {j} - 5\hat {k}$$ and $$\vec {b} = 2\hat {i} + \hat {j} - 7\hat {k}$$


Solution

Here, $$\vec{a}=2\vec{i}+2\vec{j}-5\vec{k}$$ and $$\vec{b}=2\vec{i}+\vec{j}-7\vec{k}$$
$$\vec{a}+\vec{b}=(2\vec{i}+2\vec{j}-5\vec{k})+(2\vec{i}+\vec{j}-7\vec{k})=4\vec{i}+3\vec{j}-12\vec{k}$$
$$|\vec{a}+\vec{b}|=\sqrt{4^2+3^2+(-12)^2}=\sqrt{16+9+144}=\sqrt{169}=13$$
Therefore, required unit vector = $$\dfrac{\vec{a}+\vec{b}}{|\vec{a}+\vec{b}|}=\dfrac{4\vec{i}+3\vec{j}-12\vec{k}}{13}$$

Physics

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