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

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.