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.