Browse Questions

# Find the projection of the vector $\hat i + 3\hat j + 7\hat k$ on the vector $7\hat i − \hat j + 8\hat k$ .

$\begin{array}{1 1} \frac{60 }{\sqrt{59}} \\ \frac{60 }{\sqrt{114}} \\ \frac{66 }{\sqrt{59}} \\ \frac{66 }{\sqrt{114}} \end{array}$

Toolbox:
• Projection of $\overrightarrow a$ on $\overrightarrow b$ is $\large\frac{\overrightarrow a.\overrightarrow b}{|\overrightarrow b|}$
Step 1:
Let $\overrightarrow a=\hat i+3\hat j+7\hat k$ and $\overrightarrow b=7\hat i-\hat j+8\hat k$.
Projection of $\overrightarrow a$ on $\overrightarrow b$ is $\large\frac{\overrightarrow a.\overrightarrow b}{|\overrightarrow b|}$
Step 2:
Let us find $\overrightarrow a.\overrightarrow b$
$\overrightarrow a.\overrightarrow b=(\hat i+3\hat j+7\hat k).(7\hat i-\hat j+8\hat k)$
$\qquad\;=1\times 7+3\times -1+7\times 8$
$\qquad\;= 7+ -3+56$
$\qquad\;= 60$
Step 3:
Next we have to find $\mid\overrightarrow b\mid=\sqrt{7^2+(-1)^2+8^2}$
$\qquad\qquad\qquad\qquad\quad\;\;=\sqrt{49+1+64}$
$\qquad\qquad\qquad\qquad\quad\;\;=\sqrt{114}$
Step 4:
Projection of $\overrightarrow a$ on $\overrightarrow b$ is $\large\frac{\overrightarrow a.\overrightarrow b}{|\overrightarrow b|}$
$\overrightarrow a.\overrightarrow b=60$ and $\mid \overrightarrow b\mid=\sqrt{114}$
$\Rightarrow \large\frac{60}{\sqrt{114}}$
Hence the projection of $\overrightarrow a$ on $\overrightarrow b$ is $\large\frac{60}{\sqrt{114}}$