Note: Every engineering student must have a hand calculator. If you don't have it, show the steps of getting an answer. Do not get a copy from anyone. Find the projection vector r_(21) of r_(2)=4i+2j-4k on r_(1)=14i+2j-3k vector direction. Answer: (} 4 2 -4 )=(

To find the projection vector $r_{21}$ of $r_{2}$ on $r_{1}$, we can use the formula:<br /><br />$r_{21} = \frac{r_{2} \cdot r_{1}}{\|r_{1}\|^2} \cdot r_{1}$<br /><br />where $r_{2} \cdot r_{1}$ is the dot product of $r_{2}$ and $r_{1}$, and $\|r_{1}\|$ is the magnitude of $r_{1}$.<br /><br />First, let's calculate the dot product of $r_{2}$ and $r_{1}$:<br /><br />$r_{2} \cdot r_{1} = (4)(14) + (2)(2) + (-4)(-3) = 56 + 4 + 12 = 72$<br /><br />Next, let's calculate the magnitude of $r_{1}$:<br /><br />$\|r_{1}\| = \sqrt{(14)^2 + (2)^2 + (-3)^2} = \sqrt{196 + 4 + 9} = \sqrt{209}$<br /><br />Now, we can substitute these values into the formula for $r_{21}$:<br /><br />$r_{21} = \frac{72}{209} \cdot (14i + 2j - 3k)$<br /><br />To simplify this expression, we can multiply both sides by $\frac{209}{72}$:<br /><br />$r_{21} = \frac{72}{209} \cdot (14i + 2j - 3k) \cdot \frac{209}{72}$<br /><br />$r_{21} = 14i + 2j - 3k$<br /><br />Therefore, the projection vector $r_{21}$ of $r_{2}$ on $r_{1}$ is $14i + 2j - 3k$.