Question

If two vectors a and b are perpendicular. If a magnitude 8 and b has magnitude 3 what is $\left| {a - 2b} \right|$?

Answer 1

Hint: To find the value of $\left| {\overrightarrow a - 2\overrightarrow b } \right|$, calculate the value ${\left| {\overrightarrow a - 2\overrightarrow b } \right|^2}$ of and take square root of that value.
How you can find the value of ${\left| {\overrightarrow a - 2\overrightarrow b } \right|^2}$i.e.
$ \Rightarrow {\left| {\overrightarrow a - 2\overrightarrow b } \right|^2} = \left( {\overrightarrow a - 2\overrightarrow b } \right)\left( {\overrightarrow a - 2\overrightarrow b } \right)$

Complete step by step solution: 1) Let us see what is given in the question, we have given the value of $|\overrightarrow a |\& |\overrightarrow b |$as follows
$ \Rightarrow |\overrightarrow a | = 8$ and
$ \Rightarrow |\overrightarrow b | = 3$
2) First of all, find the value of ${\left| {\overrightarrow a - 2\overrightarrow b } \right|^2}$i.e.
$ \Rightarrow {\left| {\overrightarrow a - 2\overrightarrow b } \right|^2} = \left( {\overrightarrow a - 2\overrightarrow b } \right)\left( {\overrightarrow a - 2\overrightarrow b } \right)$
$\begin{gathered}
   \Rightarrow {\left| {\overrightarrow a - 2\overrightarrow b } \right|^2} = \overrightarrow a \left( {\overrightarrow a - 2\overrightarrow b } \right) - 2\overrightarrow b \left( {\overrightarrow a - 2\overrightarrow b } \right) \\
   \Rightarrow {\left| {\overrightarrow a - 2\overrightarrow b } \right|^2} = \overrightarrow a .\overrightarrow a - \overrightarrow a .2\overrightarrow b - 2\overrightarrow b .\overrightarrow a - 2\overrightarrow b .2\overrightarrow b \\
\end{gathered} $
$ \Rightarrow {\left| {\overrightarrow a - 2\overrightarrow b } \right|^2} = \overrightarrow a .\overrightarrow a - 2\overrightarrow a ..\overrightarrow b - 2\overrightarrow a .\overrightarrow b - 4\overrightarrow b \overrightarrow b $ (as $ \Rightarrow \overrightarrow a .2\overrightarrow b = 2.\overrightarrow b \overrightarrow a = 2\overrightarrow a \overrightarrow b $)
$ \Rightarrow {\left| {\overrightarrow a - 2\overrightarrow b } \right|^2} = {\overrightarrow {\left| a \right|} ^2} - 4\overrightarrow a .\overrightarrow b - 4{\overrightarrow {\left| b \right|} ^2}$ …….(1)
3) Now, we have to find the value of $\overrightarrow a .\overrightarrow b $as given below
The dot product of a and b is given by
4) $ \Rightarrow \overrightarrow a .\overrightarrow b = |\overrightarrow a ||\overrightarrow b |\cos \theta $, where $\theta $ is the angle between a and b. we have given that a and b are perpendicular. Therefore, $\theta = {90^ \circ }$and $\cos {90^ \circ } = 0$
$ \Rightarrow \overrightarrow a .\overrightarrow b = 0$
5) Put the values of $\overrightarrow a ,\overrightarrow b \& \overrightarrow a .\overrightarrow b $ in equation (1) we get,
$ \Rightarrow {\left| {\overrightarrow a - 2\overrightarrow b } \right|^2} = {\overrightarrow {\left| a \right|} ^2} - 4\overrightarrow a .\overrightarrow b - 4{\overrightarrow {\left| b \right|} ^2}$
$\begin{gathered}
   \Rightarrow {\left| {\overrightarrow a - 2\overrightarrow b } \right|^2} = {8^2} - 4(0) - 4 \times {3^2} \\
   \Rightarrow {\left| {\overrightarrow a - 2\overrightarrow b } \right|^2} = 64 - 4 \times 9 = 64 - 36 \\
   \Rightarrow {\left| {\overrightarrow a - 2\overrightarrow b } \right|^2} = 28 \\
\end{gathered} $
Taking square root on both sides we get,
$ \Rightarrow \left| {\overrightarrow a - 2\overrightarrow b } \right| = \sqrt {28} = 2\sqrt 7 $

Therefore, the value of $\left| {\overrightarrow a - 2\overrightarrow b } \right|$ is $2\sqrt 7 $.

Note: Types of vectors are given as follows:
1) Zero or Null Vector: When starting and ending points of a vector are same is called zero or null vector.
2) Unit Vector: If the magnitude of a vector is unity then the vector is called a unit vector.
3) Free Vectors: When the initial point of a vector is not defined then those types of vectors are said to be free vectors.
4) Negative of a Vector: A vector is said to be a negative vector if the magnitude of a vector is the same as the given vector but the direction is opposite to it.
5) Like and Unlike Vectors: Unlike vectors are the vectors whose direction is opposite to each other but the direction of both is same in like vectors.