Question
If $\vec{\text{a}}=\hat{\text{i}}+2\hat{\text{j}},\ \vec{\text{b}}=\hat{\text{j}}+2\hat{\text{k}}$, write a unit vector along the vector $3\vec{\text{a}}-2\vec{\text{b}}$.
✓
Answer
Given: $\vec{\text{a}}=\hat{\text{i}}+2\hat{\text{j}},\ \vec{\text{b}}=\hat{\text{j}}+2\hat{\text{k}}$
Therefore,
$3\vec{\text{a}}-2\vec{\text{b}}=3\hat{\text{i}}+6\hat{\text{j}}-2\hat{\text{j}}-4\hat{\text{k}}$
$=3\hat{\text{i}}+4\hat{\text{j}}-4\hat{\text{k}}$
Hence, Unit vector along $3\vec{\text{a}}-2\vec{\text{b}}=\frac{3\hat{\text{i}}+4\hat{\text{j}}-4\hat{\text{k}}}{\sqrt{3^2+4^2+(-4)^2}}$
$=\frac{3\hat{\text{i}}+4\hat{\text{j}}-4\hat{\text{k}}}{\sqrt{9+16+16}}$
$=\frac{1}{\sqrt{41}}\big(3\hat{\text{i}}+4\hat{\text{j}}-4\hat{\text{k}}\big)$
Therefore,
$3\vec{\text{a}}-2\vec{\text{b}}=3\hat{\text{i}}+6\hat{\text{j}}-2\hat{\text{j}}-4\hat{\text{k}}$
$=3\hat{\text{i}}+4\hat{\text{j}}-4\hat{\text{k}}$
Hence, Unit vector along $3\vec{\text{a}}-2\vec{\text{b}}=\frac{3\hat{\text{i}}+4\hat{\text{j}}-4\hat{\text{k}}}{\sqrt{3^2+4^2+(-4)^2}}$
$=\frac{3\hat{\text{i}}+4\hat{\text{j}}-4\hat{\text{k}}}{\sqrt{9+16+16}}$
$=\frac{1}{\sqrt{41}}\big(3\hat{\text{i}}+4\hat{\text{j}}-4\hat{\text{k}}\big)$
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