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Mathematics Question on Vector Algebra

For given vectors,a⃗=2i^−j^+2k^\vec{a}=2\hat{i}-\hat{j}+2\hat{k} and b⃗=−i^+j^−k^\vec{b}=-\hat{i}+\hat{j}-\hat{k},find the unit vector in the direction of the vector a⃗+b⃗.\vec{a}+\vec{b}.

Answer

The correct answer is:12i^+12k^.\frac{1}{2}\hat{i}+\frac{1}{\sqrt{2}}\hat{k}.
The given vectors are a⃗=2i^−j^+2k^\vec{a}=2\hat{i}-\hat{j}+2\hat{k} and b⃗=−i^+j^−k^\vec{b}=-\hat{i}+\hat{j}-\hat{k}
a⃗=2i^−j^+2k^\vec{a}=2\hat{i}-\hat{j}+2\hat{k}
b⃗=−i^+j^−k^\vec{b}=-\hat{i}+\hat{j}-\hat{k}
∴a⃗+b⃗=(2−1)i^+(−1+1)j^+(2−1)k^=1i^+0j^+1k^=i^+k^∴\vec{a}+\vec{b}=(2-1)\hat{i}+(-1+1)\hat{j}+(2-1)\hat{k}=1\hat{i}+0\hat{j}+1\hat{k}=\hat{i}+\hat{k}
∣a⃗+b⃗∣=12+12=2|\vec{a}+\vec{b}|=\sqrt{1^2+1^2}=\sqrt{2}
Hence,the unit vector in the direction of (a⃗+b⃗)(\vec{a}+\vec{b}) is
(a⃗+b⃗)∣a⃗+b⃗∣=i^+k^2=12i^+12k^.\frac{(\vec{a}+\vec{b})}{|\vec{a}+\vec{b}|}=\frac{\hat{i}+\hat{k}}{\sqrt{2}}=\frac{1}{2}\hat{i}+\frac{1}{\sqrt{2}}\hat{k}.