Question

If \[\left| {\overrightarrow a } \right| = 2,\left| {\overrightarrow b } \right| = 3\] and \[\left| {2\overrightarrow a - \overrightarrow b } \right| = 5\], then \[\left| {2\overrightarrow a + \overrightarrow b } \right|\] equals
(A) 17
(B) 7
(C) 5
(D) 1

Answer 1

Hint: Here first we will use the given quantity i.e. \[\left| {2\overrightarrow a - \overrightarrow b } \right| = 5\] and square both the sides and expand the resultant quantity.
Then put the known values to and find the value of the required quantity.

Complete step-by-step answer:
It is given that:
\[\left| {2\overrightarrow a - \overrightarrow b } \right| = 5\]
Squaring both the sides we get:-
\[{\left( {\left| {2\overrightarrow a - \overrightarrow b } \right|} \right)^2} = {\left( 5 \right)^2}\]
Solving it further we get:-
\[\left| {2\overrightarrow a - \overrightarrow b } \right|.\left| {2\overrightarrow a - \overrightarrow b } \right| = 25\]
Now on multiplying we get:-
\[\left( 2 \right)\left( 2 \right)\left( {{{\left| {\overrightarrow a } \right|}^2}} \right) + \left( 1 \right)\left( 1 \right)\left( {{{\left| {\overrightarrow b } \right|}^2}} \right) - 2\left( {\overrightarrow a .\overrightarrow b } \right) - 2\left( {\overrightarrow a .\overrightarrow b } \right) = 25\]
Simplifying it further we get:-
\[4\left( {{{\left| {\overrightarrow a } \right|}^2}} \right) + \left( {{{\left| {\overrightarrow b } \right|}^2}} \right) - 4\left( {\overrightarrow a .\overrightarrow b } \right) = 25\]
Now it is given that:
\[\left| {\overrightarrow a } \right| = 2,\left| {\overrightarrow b } \right| = 3\]
Hence putting in the known values we get:-
\[4{\left( 2 \right)^2} + {\left( 3 \right)^2} - 4\left( {\overrightarrow a .\overrightarrow b } \right) = 25\]
Now solving it further we get:-
\[
  4\left( 4 \right) + 9 - 4\left( {\overrightarrow a .\overrightarrow b } \right) = 25 \\
   \Rightarrow 16 + 9 - 4\left( {\overrightarrow a .\overrightarrow b } \right) = 25 \\
   \Rightarrow 25 - 4\left( {\overrightarrow a .\overrightarrow b } \right) = 25 \\
 \]
Evaluating the value of \[4\left( {\overrightarrow a .\overrightarrow b } \right)\] we get:-
\[
  4\left( {\overrightarrow a .\overrightarrow b } \right) = 25 - 25 \\
  4\left( {\overrightarrow a .\overrightarrow b } \right) = 0...............................\left( 1 \right) \\
 \]
Now we will evaluate the value of \[\left| {2\overrightarrow a + \overrightarrow b } \right|\].
On squaring the give quantity we get:-
\[{\left| {2\overrightarrow a + \overrightarrow b } \right|^2} = \left( {2\overrightarrow a + \overrightarrow b } \right).\left( {2\overrightarrow a + \overrightarrow b } \right)\]
Solving it further we get:-
\[{\left| {2\overrightarrow a + \overrightarrow b } \right|^2} = \left( 2 \right)\left( 2 \right)\left( {{{\left| {\overrightarrow a } \right|}^2}} \right) + \left( 1 \right)\left( 1 \right)\left( {{{\left| {\overrightarrow b } \right|}^2}} \right) + 2\left( {\overrightarrow a .\overrightarrow b } \right) + 2\left( {\overrightarrow a .\overrightarrow b } \right)\]
On simplifying we get:-
\[{\left| {2\overrightarrow a + \overrightarrow b } \right|^2} = 4\left( {{{\left| {\overrightarrow a } \right|}^2}} \right) + \left( {{{\left| {\overrightarrow b } \right|}^2}} \right) + 4\left( {\overrightarrow a .\overrightarrow b } \right)\]
Now it is given that:
\[\left| {\overrightarrow a } \right| = 2,\left| {\overrightarrow b } \right| = 3\]
Hence putting in the known values we get:-
\[{\left| {2\overrightarrow a + \overrightarrow b } \right|^2} = 4{\left( 2 \right)^2} + {\left( 3 \right)^2} - 4\left( {\overrightarrow a .\overrightarrow b } \right)\]
Now putting the value from equation1 we get:-
\[{\left| {2\overrightarrow a + \overrightarrow b } \right|^2} = 4{\left( 2 \right)^2} + {\left( 3 \right)^2} - 0\]
Solving it further we get:-
\[
  {\left| {2\overrightarrow a + \overrightarrow b } \right|^2} = 4\left( 4 \right) + 9 \\
   \Rightarrow {\left| {2\overrightarrow a + \overrightarrow b } \right|^2} = 16 + 9 \\
   \Rightarrow {\left| {2\overrightarrow a + \overrightarrow b } \right|^2} = 25 \\
 \]
Now taking square root of both the sides we get:-
\[\sqrt {{{\left| {2\overrightarrow a + \overrightarrow b } \right|}^2}} = \sqrt {25} \]
Evaluating it further we get:-
\[\left| {2\overrightarrow a + \overrightarrow b } \right| = \pm 5\]
But since we know that the modulus of any quantity is always positive
Hence we will consider the positive value
Therefore,
\[\left| {2\overrightarrow a + \overrightarrow b } \right| = 5\]
Hence option C is the correct option.

Note: Students should keep in mind that modulus of any quantity is always positive and also when we take squares of any vector quantity then they are multiplied with each other using dot product.