Question

If $\mathop a\limits^ \wedge$ , $\mathop b\limits^ \wedge$ and $\mathop c\limits^ \wedge$ are unit vectors satisfying ${\left| {\mathop a\limits^ \wedge - \mathop b\limits^ \wedge } \right|^2} + {\left| {\mathop b\limits^ \wedge - \mathop c\limits^ \wedge } \right|^2} + {\left| {\mathop c\limits^ \wedge - \mathop a\limits^ \wedge } \right|^2} = 9$ then $\left| {2\mathop a\limits^ \wedge + 5\mathop b\limits^ \wedge + 5\mathop c\limits^ \wedge } \right|$ is
A 3
B 4
C 5
D 6

Answer 1

Hint : A vector that has a magnitude of 1 is a unit vector, any vector can become a unit vector by dividing it by the magnitude of the given vector, and if $\mathop a\limits^ \wedge$ , $\mathop b\limits^ \wedge$ and $\mathop c\limits^ \wedge$ are unit vectors satisfying ${\left| {\mathop a\limits^ \wedge - \mathop b\limits^ \wedge } \right|^2} + {\left| {\mathop b\limits^ \wedge - \mathop c\limits^ \wedge } \right|^2} + {\left| {\mathop c\limits^ \wedge - \mathop a\limits^ \wedge } \right|^2} = 9$ , then we can expand the terms using the formula of ${\left( {a - b} \right)^2}$ and then simplify for $\left| {2\mathop a\limits^ \wedge + 5\mathop b\limits^ \wedge + 5\mathop c\limits^ \wedge } \right|$ .
Formula used:
${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab$
${\left( {b - c} \right)^2} = {b^2} + {c^2} - 2bc$
${\left( {c - a} \right)^2} = {c^2} + {a^2} - 2ca$
In which a, b, c are unit vectors.

Complete step-by-step answer :
Let us write the given data:
${\left| {\mathop a\limits^ \wedge - \mathop b\limits^ \wedge } \right|^2} + {\left| {\mathop b\limits^ \wedge - \mathop c\limits^ \wedge } \right|^2} + {\left| {\mathop c\limits^ \wedge - \mathop a\limits^ \wedge } \right|^2} = 9$ ${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab$
We can see that the given vector is of the form ${\left( {a - b} \right)^2}$ , hence let us apply the expansion of that formula as
We know that,
, ${\left( {b - c} \right)^2} = {b^2} + {c^2} - 2bc$ and ${\left( {c - a} \right)^2} = {c^2} + {a^2} - 2ca$
Hence, applying to the given vectors:
${\left| {\mathop a\limits^ \wedge - \mathop b\limits^ \wedge } \right|^2} + {\left| {\mathop b\limits^ \wedge - \mathop c\limits^ \wedge } \right|^2} + {\left| {\mathop c\limits^ \wedge - \mathop a\limits^ \wedge } \right|^2} = 9$
$\Rightarrow$ ${\left| {\mathop a\limits^ \wedge } \right|^2} + {\left| {\mathop b\limits^ \wedge } \right|^2} - 2\mathop a\limits^ \wedge \cdot \mathop b\limits^ \wedge + {\left| {\mathop b\limits^ \wedge } \right|^2} + {\left| {\mathop c\limits^ \wedge } \right|^2} - 2\mathop b\limits^ \wedge \cdot \mathop c\limits^ \wedge + {\left| {\mathop c\limits^ \wedge } \right|^2} + {\left| {\mathop a\limits^ \wedge } \right|^2} - 2\mathop c\limits^ \wedge \cdot \mathop a\limits^ \wedge = 9$
Now taking the common terms we get the equation as:
$2\left( {{{\left| {\mathop a\limits^ \wedge } \right|}^2} + {{\left| {\mathop b\limits^ \wedge } \right|}^2} + {{\left| {\mathop c\limits^ \wedge } \right|}^2}} \right) - 2\left( {\mathop a\limits^ \wedge \cdot \mathop b\limits^ \wedge + \mathop b\limits^ \wedge \cdot \mathop c\limits^ \wedge + \mathop c\limits^ \wedge \cdot \mathop a\limits^ \wedge } \right) = 9$
$3\left( {{{\left| {\mathop a\limits^ \wedge } \right|}^2} + {{\left| {\mathop b\limits^ \wedge } \right|}^2} + {{\left| {\mathop c\limits^ \wedge } \right|}^2}} \right) - \left( {{{\left| {\mathop a\limits^ \wedge } \right|}^2} + {{\left| {\mathop b\limits^ \wedge } \right|}^2} + {{\left| {\mathop c\limits^ \wedge } \right|}^2} + 2\left( {\mathop a\limits^ \wedge \cdot \mathop b\limits^ \wedge + \mathop b\limits^ \wedge \cdot \mathop c\limits^ \wedge + \mathop c\limits^ \wedge \cdot \mathop a\limits^ \wedge } \right) = 9} \right)$
Since, a, b, c are unit vectors we get
$3\left( 3 \right) - {\left| {\mathop a\limits^ \wedge + \mathop b\limits^ \wedge + \mathop c\limits^ \wedge } \right|^2} = 9$
$9 - {\left| {\mathop a\limits^ \wedge + \mathop b\limits^ \wedge + \mathop c\limits^ \wedge } \right|^2} = 9$
Hence, we get
$\left| {\mathop a\limits^ \wedge + \mathop b\limits^ \wedge + \mathop c\limits^ \wedge } \right| = 0$
$\Rightarrow$ $\mathop a\limits^ \wedge + \mathop b\limits^ \wedge + \mathop c\limits^ \wedge = 0$ and $\mathop b\limits^ \wedge + \mathop c\limits^ \wedge = - \mathop a\limits^ \wedge$
Hence, we get
$\left( {\mathop a\limits^ \wedge + \mathop b\limits^ \wedge + \mathop c\limits^ \wedge } \right) = 0$
Now, we need to find the vector $\left| {2\mathop a\limits^ \wedge + 5\mathop b\limits^ \wedge + 5\mathop c\limits^ \wedge } \right|$
We can combine the common terms from the given vector as:
$\left| {2\mathop a\limits^ \wedge + 5\mathop b\limits^ \wedge + 5\mathop c\limits^ \wedge } \right| = \left| {2\mathop a\limits^ \wedge + 5\left( {\mathop b\limits^ \wedge + \mathop c\limits^ \wedge } \right)} \right|$
We know that, since $\mathop b\limits^ \wedge + \mathop c\limits^ \wedge = - \mathop a\limits^ \wedge$ we get
= $\left| {2\mathop a\limits^ \wedge + 5\left( { - \mathop a\limits^ \wedge } \right)} \right|$
$\Rightarrow$ $\left| { - 3\mathop a\limits^ \wedge } \right| = 3\left| {\mathop a\limits^ \wedge } \right|$
= $3 \times 1$
= $3$
Therefore, option A is the right answer.
So, the correct answer is “Option A”.

Note : A vector is a quantity that has both magnitudes, as well as direction. Unit vectors are usually determined to form the base of a vector space. Every vector in the space can be expressed as a linear combination of unit vectors.
To find a unit vector with the same direction as a given vector, we divide the vector by its magnitude and these unit vectors are commonly used to indicate direction, with a scalar coefficient providing the magnitude. A unit vector contains directional information and if you multiply a positive scalar by a unit vector, then you produce a vector with magnitude equal to that scalar in the direction of the unit vector.