Question

Find the magnitude of the vector which starts at the point $2\hat i + \hat j - 3\hat k$ and ends at $4\hat i - \hat j - \hat k$.

Answer 1

$Hint:$ In this question we will use one of the types of vectors i.e. position vector which is $\overrightarrow {ab} = \overrightarrow b - \overrightarrow a$ as it used to specify the positions of the vector. Where $\overrightarrow b$ is the ending point and $\overrightarrow a$ is the starting point of a vector .
Complete step-by-step solution -
According to the question two points where the vector starts and ends are given i.e. $2\hat i + \hat j - 3\hat k$ and $4\hat i - \hat j - \hat k$ respectively.
Now, let vector $\overrightarrow a = 2\hat i + \hat j - 3\hat k$
$\overrightarrow b = 4\hat i - \hat j - \hat k$
Use the Position Vector,
Hence $\overrightarrow {ab} = \overrightarrow b - \overrightarrow a$
$= 4\hat i - \hat j - \hat k - 2\hat j - \hat j + 3\hat k \\\ = 2\hat i - 2\hat j + 2\hat k \\\$
Now, $\left| {\overrightarrow {ab} } \right| = \sqrt {{{\left( 2 \right)}^2} + {{\left( { - 2} \right)}^2} + {{\left( 2 \right)}^2}}$
$= \sqrt {4 + 4 + 4} \\\ = \sqrt {12} \\\ = 2\sqrt 3 \\\$
Hence the magnitude of the vector is $2\sqrt 3$.
$Note$ : In such types of questions where the magnitude of the vector has to find there we use the position vector i.e. if vector $\overrightarrow {OA}$ is used to specify the position of a point $A$ relative to another point $O$. This $\overrightarrow {OA}$ is called the position vector of $A$ referred to $O$ as an origin. These concepts will help in solving vector questions so it is advisable to remember these concepts.