Question

If \[a=-i+2j-k,b=i+j-3k,c=-4i-k,\] then \[a\times \left( b\times c \right)+\left( a.b \right)c\] will be
\[\left( a \right)5i+5j-15k\]
\[\left( b \right)0\]
\[\left( c \right)12j+4k\]
\[\left( d \right)-3i+6j-3k\]

Answer 1

Hint: We are having three vectors a, b and c. We are asked to find the value of \[a\times \left( b\times c \right)+\left( a.b \right)c.\] We will use the formula \[a\times \left( b\times c \right)=\left( a.c \right)b-\left( a.b \right)c\] to find the value of \[a\times \left( b\times c \right)+\left( a.b \right)c.\] Then we will use the dot product formula \[{{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}+{{c}_{1}}{{c}_{2}}\] to find the dot product of a and c. Lastly, we will simplify to get our required answer.

Complete step by step answer:
We are given that we have three vectors
\[\begin{align}
  & a=-i+2j-k \\
 & b=i+j-3k \\
 & c=-4i-k \\
\end{align}\]
We are asked to find the value of \[a\times \left( b\times c \right)+\left( a.b \right)c.\] We know that the triple vector product of the vector say x, y and z are given as
\[x\times \left( y\times z \right)=\left( x.z \right)y-\left( x.y \right)z\]
So, from this, we get,
\[x\times \left( y\times z \right)+\left( x.y \right)z=\left( x.z \right)y\]
Now, consider x as a, y as b and z as c, so we will get,
\[a\times \left( b\times c \right)+\left( a.b \right)c=\left( a.c \right)b\]
So to find our required answer, we will first find the dot product of a and b. We know that the dot product of \[X={{a}_{1}}i+{{b}_{1}}j+{{c}_{1}}k\] and \[Y={{a}_{2}}i+{{b}_{2}}j+{{c}_{2}}k\] is given as
\[X.Y={{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}+{{c}_{1}}{{c}_{2}}\]
So for \[a=-i+2j-k;c=-4i-k\]
We can compute the dot product as below,
\[a.c=\left( -i+2j-k \right)\left( -4i-k \right)\]
Now, multiplying the corresponding coefficients and adding them, we have
\[\Rightarrow a.c=-1\times -4+2\times 0+\left( -1 \right)\left( -1 \right)\]
Simplifying terms, we get
\[\Rightarrow a.c=4+0+1\]
\[\Rightarrow a.c=5\]
Now putting it in \[a\times \left( b\times c \right)+\left( a.b \right)c=\left( a.c \right)b\] we get,
\[\Rightarrow a\times \left( b\times c \right)+\left( a.b \right)c=5\left( i+j-3k \right)\]
\[\Rightarrow a\times \left( b\times c \right)+\left( a.b \right)c=5i+5j-15k\]

So, the correct answer is “Option A”.

Note: We do not have to do the long way that is by first finding the triple product of \[a\times \left( b\times c \right)\] and then solving (a.b)c and then at last add and then we just have to use the expansion formula of the triple product of the vector. While applying the dot product, always remember to multiply only the corresponding times and not the others.