Question

Find a vector of magnitude 5 units, and parallel to the resultant of the vectors
$\overrightarrow a = 2\hat i + 3\hat j - \hat k$ and \[\overrightarrow b = \hat i - 2\hat j + \hat k\].

Answer 1

Hint: First find the resultant vector of the two vectors. After that find the modulus of the resultant vector. Then, use the formula $\overrightarrow x = 5\left( {\dfrac{{\overrightarrow a + \overrightarrow b }}{{\left| {\overrightarrow a + \overrightarrow b } \right|}}} \right)$ to find the vector parallel to the resultant of the vectors and has a magnitude of 5 units. Then, rationalize the denominator of the output and simplify the terms to get the desired result.

Complete step-by-step answer:
Given: - $\overrightarrow a = 2\hat i + 3\hat j - \hat k$
\[\overrightarrow b = \hat i - 2\hat j + \hat k\]
Now we have to find out the vector which is parallel to the resultant of the vectors and has a magnitude of 5 units.
As we know when we divide the vector with its modulus then the vector converts into the unit vector and when we multiply by 5 with this unit vector the vector converts into the vector which has magnitude 5 units.
Let the parallel vector to the given vector be,
$ \Rightarrow \overrightarrow x = 5\left( {\dfrac{{\overrightarrow a + \overrightarrow b }}{{\left|
{\overrightarrow a + \overrightarrow b } \right|}}} \right)$ …….... (1)
The resultant of the vectors is,
\[ \Rightarrow \overrightarrow a + \overrightarrow b = \left( {2\hat i + 3\hat j - \hat k} \right) + \left(
{\hat i - 2\hat j + \hat k} \right)\]
Group the like terms on the right side,
\[ \Rightarrow \overrightarrow a + \overrightarrow b = \left( {2 + 1} \right)\hat i + \left( {3 - 2}
\right)\hat j + \left( { - 1 + 1} \right)\hat k\]
Add the terms,
\[ \Rightarrow \overrightarrow a + \overrightarrow b = 3\hat i + 1\hat j + 0\hat k\]
……..... (2)
Now, the modulus of the vector $\overrightarrow p = x\hat i + y\hat j + z\hat k$ is given by,
$\left| {\overrightarrow p } \right| = \sqrt {{x^2} + {y^2} + {z^2}} $
Use this property to calculate to find the modulus of the resultant vectors.
Substitute the values,
$ \Rightarrow \left| {\overrightarrow a + \overrightarrow b } \right| = \sqrt {{3^2} + {1^2} + {0^2}} $
Square the terms and add,
$ \Rightarrow \left| {\overrightarrow a + \overrightarrow b } \right| = \sqrt {10} $
………... (3)
Substitute the values from equation (2) and (3) in equation (1),
$ \Rightarrow \overrightarrow x = 5\left( {\dfrac{{3\hat i + \hat j}}{{\sqrt {10} }}} \right)$
Multiply the numerator and denominator by $\sqrt {10} $ to rationalize the denominator,
$ \Rightarrow \overrightarrow x = 5\left( {\dfrac{{3\hat i + \hat j}}{{\sqrt {10} }}} \right) \times
\dfrac{{\sqrt {10} }}{{\sqrt {10} }}$
Simplify the terms,
$ \Rightarrow \overrightarrow x = \dfrac{{5\sqrt {10} }}{{10}}\left( {3\hat i + \hat j} \right)$
Cancel out the common factors,
$ \Rightarrow \overrightarrow x = \dfrac{{\sqrt {10} }}{2}\left( {3\hat i + \hat j} \right)$
Open the brackets and multiply,
\[\therefore \overrightarrow x = \dfrac{{3\sqrt {10} }}{2}\hat i + \dfrac{{\sqrt {10} }}{2}\hat j\]

Hence, the required vector which is parallel to the resultant of the vectors and has magnitude 5 units is \[\dfrac{{3\sqrt {10} }}{2}\hat i + \dfrac{{\sqrt {10} }}{2}\hat j\].

Note: Whenever we face such type of problems the key concept is simply to take out the unit vector of the given vector as it will be in the direction of the given vector then multiply it with the scalar constant to obtain the desired magnitude vector.