Question

If A=2i+j+k, B=i-2j+2k, and C=3i-4j+2k, find the projection of A+C in the direction of B.

Answer 1

Question 1.

If $A=2i+j+k$, $B=i-2j+2k$, and $C=3i-4j+2k$, find the projection of $A+C$ in the direction of $B$.

Solution. First of all find $A+C$. We have

$A+C=(2i+j+k)+(3i-4j+2k)=(2+3)i+(1-4)j+(1+2)k=5i-3j+3k.$

The projection can be calculated by the formula

$pr_{B}(A+C)=\frac{B(A+C)}{|B|^{2}}B,$

where $B(A+C)$ is the scalar product of $B$ and $A+C$, $|B|^{2}$ is the square of the absolute value of $B$. Find these values:

$B(A+C)=(i-2j+2k)(5i-3j+3k)=1\cdot 5+(-2)\cdot(-3)+2\cdot 3=5+6+6=17,$

and

$|B|^{2}=(i-2j+2k)^{2}=1^{2}+(-2)^{2}+2^{2}=1+4+4=9.$

Thus,

$pr_{B}(A+C)=\frac{17}{9}(i-2j+2k)=\frac{17}{9}i-\frac{34}{9}j+\frac{34}{9}k.$

Answer: $\frac{17}{9}i-\frac{34}{9}j+\frac{34}{9}k$. $\square$