Question

Let $\overrightarrow{a}=2\hat{i}+3\hat{i}+2\hat{k}$ and $\overrightarrow{b}=\hat{i}+2\hat{i}+\hat{k}$.
Find the projection of
1.$\overrightarrow{a}$ on $\overrightarrow{b}$
2. $\overrightarrow{b}$ on $\overrightarrow{a}$

Answer 1

Consider given the given vectors,$$\overrightarrow{a}=2\hat{i}+3\hat{j}+2\hat{k}$$ and $\overrightarrow{b}=\hat{i}+2\hat{j}+\hat{k}$

Now, projection of $\overrightarrow{a}$ on $\overrightarrow{b}$ $=\frac{\overrightarrow{a}.\overrightarrow{b}}{\left|\overrightarrow{b}\right|}=\frac{(2\hat{i}+3\hat{j}+2\hat{k}).(\hat{i}+2\hat{j}+\hat{k})}{|\hat{i}+2\hat{j}+\hat{k}|}$

$=\frac{(2\hat{i}+3\hat{j}+2\hat{k}).(\hat{i}+2\hat{j}+\hat{k})}{|\hat{i}+2\hat{j}+\hat{k}|}=\frac{2.1+3.2+2.1}{\sqrt{1^2+2^2+1^2}}$

$=\frac{10}{\sqrt{6}}$

Projection of $\overrightarrow{b}$ on $\overrightarrow{a}$,

$=\frac{(\hat{i}+2\hat{j}+\hat{k}).(2\hat{i}+3\hat{j}+2\hat{k})}{|2\hat{i}+3\hat{j}+2\hat{k}|}=\frac{1.2+2.3+1.2}{\sqrt{2^2+3^2+2^2}}=\frac{10}{\sqrt{17}}$

Hence, this is the answer.