Question

The coordinates of head and tail of a vector are $\left( {2,1,0} \right)$ and $\left( { - 4,2, - 3} \right)$ respectively. Find the magnitude of the vector.
A. $\sqrt {23} \,{\text{units}}$
B. $\sqrt {46} \,{\text{units}}$
C. $\sqrt {84} \,{\text{units}}$
D. $\sqrt {12} \,{\text{units}}$

Answer 1

If coordinate of a point in three-dimensional space is given as
$\left( {x,y,z} \right)$
Then we can write the position vector of this point in vector form as
$x\,\hat i + y\,\hat j + z\,\hat k$
If we have two vectors
$\overrightarrow P = {x_1}\hat i + {y_1}\hat j + {z_1}\hat k$
And
$\overrightarrow Q = {x_2}\hat i + {y_2}\hat j + {z_2}\hat k$
Then
$\overrightarrow {PQ}$ is given as
$\overrightarrow {PQ} = \left( {{x_2} - {x_1}} \right)\hat i + \left( {{y_2} - {y_1}} \right)\hat j + \left( {{z_2} - {z_1}} \right)\hat k$
For a vector $x\,\hat i + y\,\hat j + z\,\hat k$the magnitude is given as
$\sqrt {{x^2} + {y^2} + {z^2}}$

Complete step by step answer:
A quantity that has both magnitude and direction is called a vector. For example, displacement, velocity etc. are vector quantities.
If coordinate of a point in three-dimensional space is given as
$\left( {x,y,z} \right)$
Then we can write the position vector of this point in vector form as
$x\,\hat i + y\,\hat j + z\,\hat k$
Where $x$ is the component in x direction, $y$ is the component in y direction and $z$ is the component in z direction. $\hat i$ denotes the unit vector in x direction $\hat j$ denotes the unit vector in y direction and $\hat k$ denotes unit vector z direction.
Now let us express the position vector of given points in vector form.
Let head be the point A and tail be the point B
Position vector of head can be written as
$\overrightarrow A = 2\hat i + 1\hat j + 0\hat k$
Position vector of tail can be written as
$\overrightarrow B = - 4\hat i + 2\hat j + - 3\hat k$
Now the vector representing the line from head to tail is given by subtracting these two vectors.
Let this line be AB
If we have two vectors
$\overrightarrow P = {x_1}\hat i + {y_1}\hat j + {z_1}\hat k$
And
$\overrightarrow Q = {x_2}\hat i + {y_2}\hat j + {z_2}\hat k$
Then
$\overrightarrow {PQ}$ is given as
$\overrightarrow {PQ} = \left( {{x_2} - {x_1}} \right)\hat i + \left( {{y_2} - {y_1}} \right)\hat j + \left( {{z_2} - {z_1}} \right)\hat k$
Using this we can write $\overrightarrow {AB}$ as
$\overrightarrow {AB} = \left( { - 4 - 2} \right)\hat i + \left( {2 - 1} \right)\hat j + \left( { - 3 - 0} \right)\hat k \\\ = - 6\hat i + 1\hat j - 3\hat k \\\$
For a vector $x\,\hat i + y\,\hat j + z\,\hat k$ the magnitude is given as
$\sqrt {{x^2} + {y^2} + {z^2}}$
Therefore, the magnitude of vector $\overrightarrow {AB}$ is given as
$\sqrt {{{\left( { - 6} \right)}^2} + {1^2} + {{\left( { - 3} \right)}^2}} = \sqrt {46}$units
So, the correct answer is option B.

Note: Formula to remember-
If we have two vectors
$\overrightarrow P = {x_1}\hat i + {y_1}\hat j + {z_1}\hat k$
And
$\overrightarrow Q = {x_2}\hat i + {y_2}\hat j + {z_2}\hat k$
Then
$\overrightarrow {PQ}$ is given as
$\overrightarrow {PQ} = \left( {{x_2} - {x_1}} \right)\hat i + \left( {{y_2} - {y_1}} \right)\hat j + \left( {{z_2} - {z_1}} \right)\hat k$
For a vector $x\,\hat i + y\,\hat j + z\,\hat k$ the magnitude is given as
$\sqrt {{x^2} + {y^2} + {z^2}}$