Question

If the equation of a line through a point a and parallel to vector b is\[r = a + tb\], where t is a parameter, then its perpendicular distance from the point c is [MP PET\[1998\]]
A) \[|(c - b) \times a| \div |a|\]
B) \[\;|(c - a) \times b| \div |b|\]
C) \[\;\;\;\;\;\;|(a - b) \times c| \div |c|\]
D) \[\;|(a - b) \times c| \div |a + c|\]

Answer 1

Hint: in this question we have to find the perpendicular distance of a given line from given point. It can be found by simply taking a point on a line and then find the position vector of point on line and given point. Then in order to find distance take modulus of position vector.



Formula Used:Equation of line is given by
\[r = a + tb\]
Where
a is a point through which line passes
b is a vector parallel to line
t is a constant
Formula for position vector is given by
\[\overrightarrow {AB} = Position{\rm{ }}vector{\rm{ }}of{\rm{ }}B{\rm{ }}-{\rm{ }}position{\rm{ }}vector{\rm{ }}of{\rm{ }}A\]



Complete step by step solution:Equation of line is given by
\[r = a + tb\]
Let some point A on a line
This line is passing through point B (say) having position vector c
\[\overrightarrow {AB} = Position{\rm{ }}vector{\rm{ }}of{\rm{ }}B{\rm{ }}-{\rm{ }}position{\rm{ }}vector{\rm{ }}of{\rm{ }}A\]
\[\overrightarrow {AB} = c - (a + tb)\]
As we know that \[\overrightarrow {AB} \]$is perpendicular to the line which is parallel to vector b
Therefore
$\[\overrightarrow {AB} .\overrightarrow b = 0\]
\[(c - (a + tb)).b = 0\]
\[cb - ab - t{b^2} = 0\]
\[t = \dfrac{{b(c - a)}}{{{b^2}}}\]
Substitute value of t in equation \[\overrightarrow {AB} = c - (a + tb)\]
\[\overrightarrow {AB} = c - (a + (\dfrac{{b(c - a)}}{{{b^2}}})b)\]$

Now distance of point c from given line is given by
$\[d = \left| {\overrightarrow {AB} } \right|\]
\[d = \left| {c - (a + (\dfrac{{b(c - a)}}{{{b^2}}})b)} \right|\]
\[d = \left| {c - a - \dfrac{{(c - a)b.b}}{{{b^2}}}} \right|\]
\[d = \left| {\dfrac{{(c - a)b.b - (c - a)b \times b}}{{{b^2}}}} \right|\]
\[d = \left| {\dfrac{{0 - (c - a)b \times b}}{{{b^2}}}} \right|\] (Because b.b=0)
\[d = \dfrac{{\left| b \right|(c - a) \times b}}{{\left| {{b^2}} \right|}}\]
\[d = \dfrac{{\left| b \right|\left| {(c - a) \times b} \right|\sin ({{90}^0})}}{{\left| {{b^2}} \right|}}\] (Because\[b \bot (c - a) \times b)\])
\[d = \dfrac{{\left| {(c - a) \times b} \right|}}{{\left| {{b^{}}} \right|}}\]$



Option ‘B’ is correct



Note: This is a standard result. Use it where question asked to find length of the perpendicular from the point to the line.
Always take a point on a line whenever question asked to find the perpendicular distance of a line from a point.