Question

Find the sum of following vectors : $a=i-3k$, $b=2j-k$ , $c=2i-3j+2k$

Answer 1

In the question, we are given three vectors in terms of $i,j,k$ and we are asked to find the sum of these three vectors, so we need to add all the $i$ components of vectors separately then $j$ component and finally the $k$ components of all three vectors

Complete step-by-step solution:
We are given three different vectors as $a=i-3k$, $b=2j-k$ , $c=2i-3j+2k$ and we need to find their sum so using formula
$a=xi+yj+zk$ and $b={{x}_{1}}i+{{y}_{1}}j+{{z}_{1}}k$ then vector addition of a and b is
$a+b=(xi+yj+zk)+({{x}_{1}}i+{{y}_{1}}j+{{z}_{1}}k)$
Now while adding two vectors we always add $i$ components of vectors separately then $j$ component and finally the $k$ components separately
So, vector addition of $a+b$ will be equal to $a+b=(x+{{x}_{1}})i+(y+{{y}_{1}})j+(z+{{z}_{1}})k$
Similar formula can be applying for any number of vectors also
So using this property in our question as vectors are $a=i-3k$, $b=2j-k$, $c=2i-3j+2k$
So, vector addition $a+b+c$ will be equals to
$a+b+c=(i-3k)+(2j-k)+(2i-3j+2k)$
$a+b+c=(1+2)i+(2-3)j+(-3-1+2)k$
Which on solving equals to $a+b+c=3i-j-2k$
Hence vector sum of three vectors given as $a=i-3k$, $b=2j-k$ , $c=2i-3j+2k$
Is equals to $a+b+c=3i-j-2k$

Note: Most of the students have one doubt related to vector addition that why vector addition is different than normal number addition, so the answer is that because vectors have direction, here I, j, and k represents x,y, and z-direction respectively so while adding or subtracting them we can add or subtract the value of the same direction only. Sometimes a line is given in the question that two vectors are perpendicular, so here we can apply one property directly that dot product of both vectors will be zero. For example dot product of $a=i-3k$ and $b=2j-k$ will be $a.b=(i-3k).(2j-k)$
$a.b=(i-3k).(2j-k)=0i+0j+3k=3k$