Question

Solve the given question with the given condition in the question:

$$let\,\lambda = a \times (b + c),\,\mu = b \times (c + a),\,V = c \times (a + b)\,then\, \\\ 1)\,\lambda + \mu = V\\\ 2)\lambda ,\mu ,V\,are\,coplanar\\\ 3)\,\lambda + V = 2\mu \\\ 4)None \\\$$

Here all the variables used are in vector form.

Answer 1

Hint : The given question simplified the given vector values and asked to find the relation between these vectors, here we have to put the values of the vector in the given relation and then check for the correctness of the relation given, to prove that which relation is correct or not correct, and to solve we have to use the vector formulas.

Complete step by step solution:
The given question describes three vector quantities, which are as:

$$\Rightarrow \,\lambda = a \times (b + c) \\\ \Rightarrow \,\mu = b \times (c + a) \\\ \Rightarrow \,V = c \times (a + b) \;$$

Now first we have to check for the first relation given in the question which is:
$\Rightarrow \,\lambda + \mu = V$
Now here we have to put the values of all the three vectors into the relation given and then check that equation is satisfying the relation or not, if the relation is correct the left hand side equation value should be equal to right hand side value, on solving we get:

$$\Rightarrow \,\lambda + \mu = V \\\ \Rightarrow a \times (b + c) + b \times (c + a) = c \times (a + b)\, \\\ \Rightarrow a \times b + a \times c + b \times c + b \times a = c \times a + c \times b \;$$

No further simplification is possible and the values around the equal sign are not equal hence the given relation does not satisfy the given vectors.

Now for second relation:
$\Rightarrow \lambda ,\mu ,V\,are\,coplanar$
On solving we get:

$$\, \Rightarrow \lambda ,\mu ,V\,are\,coplanar \\\ \Rightarrow For\,ve\operatorname{c} tors\,to\,be\,coplanar\,their\,scalar\,triple\,product\,should\,be\,zero: \\\ \Rightarrow i.e.\,\lambda .[\mu \times V] = 0 \\\ \Rightarrow here\,the\,triple\,product\,will\,not\,be\,zero,hence\,vectors\,are\,non\,coplanar \;$$

Now for third relation:

$$\Rightarrow \,\lambda + V = 2\mu \\\ \Rightarrow a \times (b + c) + b \times (c + a) = 2[c \times (a + b)] \, \\\ \Rightarrow a \times b + a \times c + b \times c + b \times a = 2(c \times a) + 2(c \times b) \;$$

No further simplification is possible and the values around the equal sign is not equal hence the given relation does not satisfy the given vectors.
Hence the fourth option none is correct.
So, the correct answer is “Option D”.

Note : The given question can be solved by graphical method also but here the vector given is in the variable form, not in the constant form, hence the given question needs to be solved by putting the values in the relation only.