Let \[\overline a \] , \[\overline b \] , \[\overline c \] be vectors of length 3, 4, 5 respectively. Let \[\overline a \] be the perpendicular to \[\overline b + \overline c \], \[\overline b \] to \[\overline c + \overline a \], and \[\overline c \] to \[\overline a + \overline b \]. Then \[\left| {\overline a + \overline b + \overline c } \right|\] is equal to:
A.\[2\sqrt 5 \]
B.\[2\sqrt 2 \]
C.\[10\sqrt 5 \]
D.\[5\sqrt 2 \]

Hint: Here we need to find the value of the magnitude of sum of three vectors. If two vectors are perpendicular to each other, then the dot product of these two vectors is equal to zero. We will use this property to find the value of the sum of vectors. We will find the square of the given expression and substitute the values of vectors to get the required answer.

Complete step-by-step answer:
Here we need to find the value of the magnitude of sum of three vectors i.e. \[\left| {\overline a + \overline b + \overline c } \right|\].
It is given that the magnitude of vector \[\overline a \] is equal to 3, magnitude of the vector \[\overline b \] is equal to 4 and magnitude of the vector \[\overline c \] is equal to 5.
It is given that \[\overline a \] be the perpendicular to \[\overline b + \overline c \].
We know that if two vectors are perpendicular to each other, then the dot product of these two vectors is equal to zero.
So, using this property, we get
\[ \Rightarrow \overline a \cdot \left( {\overline b + \overline c } \right) = 0\]
On further simplification, we get
\[ \Rightarrow \overline a \cdot \overline b + \overline a \cdot \overline c = 0\] ……….. \[\left( 1 \right)\]
It is given that \[\overline b \] be the perpendicular to \[\overline c + \overline a \].
Similarly, using the property again, we get
\[\overline b \cdot \left( {\overline c + \overline a } \right) = 0\]
On further simplification, we get
\[ \Rightarrow \overline b \cdot \overline a + \overline b \cdot \overline a = 0\] ……….. \[\left( 2 \right)\]
 It is given that \[\overline c \] be the perpendicular to \[\overline a + \overline b \].
Similarly, using the property again, we get
\[\overline a \cdot \left( {\overline b + \overline c } \right) = 0\]
On further simplification, we get
\[ \Rightarrow \overline a \cdot \overline b + \overline a \cdot \overline c = 0\] ……….. \[\left( 3 \right)\].
Now, we will add equation 1, equation 2 and equation 3. Therefore, we get
\[\overline a \cdot \overline b + \overline a \cdot \overline c + \overline b \cdot \overline a + \overline b \cdot \overline a + \overline a \cdot \overline b + \overline a \cdot \overline c = 0\]
On adding the like terms, we get
\[ \Rightarrow 2\left( {\overline a \cdot \overline b + \overline b \cdot \overline c + \overline a \cdot \overline c } \right) = 0\] ……….. \[\left( 4 \right)\]
We know that
\[{\left| {\overline a + \overline b + \overline c } \right|^2} = {\left| {\overline a } \right|^2} + {\left| {\overline b } \right|^2} + {\left| {\overline c } \right|^2} + 2\left( {\overline a \cdot \overline b + \overline b \cdot \overline c + \overline a \cdot \overline c } \right)\]
Substituting the value of the known vectors, we get
\[ \Rightarrow {\left| {\overline a + \overline b + \overline c } \right|^2} = {3^2} + {4^2} + {5^2} + 0\]
Applying the exponents on the bases, we get
\[ \Rightarrow {\left| {\overline a + \overline b + \overline c } \right|^2} = 9 + 16 + 25\]
On adding the numbers, we get
\[ \Rightarrow {\left| {\overline a + \overline b + \overline c } \right|^2} = 50\]
Taking square root on both sides, we get
\[ \Rightarrow \sqrt {{{\left| {\overline a + \overline b + \overline c } \right|}^2}} = \sqrt {50} \]
On further simplification, we get
\[ \Rightarrow \left| {\overline a + \overline b + \overline c } \right| = 5\sqrt 2 \]
Hence, the correct option is option D.

Note: To solve this question, we need to know the definition of vectors and the property of vectors. A vector is defined as the quantity that has direction and has magnitude both. If two vectors are parallel to each other, then the cross product of two vectors is zero. Vectors are different from scalar quantity as the scalar quantity has only magnitude and doesn’t have any direction. Examples of vector quantities are velocity, acceleration, etc. Examples of scalar quantities are mass, temperature, etc.