Question

The projection of the line segment joining (2,0,-3) and (5,-1,2) on a straight line whose direction ratios are 2,4,4 is
a) $\dfrac{{11}}{6}$
b) $\dfrac{{10}}{3}$
c) $\dfrac{{13}}{3}$
d) $\dfrac{{13}}{6}$
e) $\dfrac{{11}}{3}$

Answer 1

Hint: We are given the direction ratios of a line and also two points through which another line passes. We are asked to find the projection of one line on another. The simplest method of finding projection is by using the projection formula. We will use that but before using it we have to convert the lines to their corresponding vector form. For doing so we will use the formulas given below.
Formula used:
1) The vector representation of a line with direction ratios a, b and c is $a\hat i + b\hat j + c\hat k$ .
2) If the line passes through two points $({x_1},{y_1},{z_1})$ and $({x_2},{y_2},{z_2})$ then the vector representation of that line is given by $({x_2} - {x_1})\hat i + ({y_2} - {y_1})\hat j + ({z_2} - {z_1})\hat k$
3) The projection of a vector ‘a’ on another vector ‘b’ is given by the formula
$\dfrac{{\vec a \cdot \vec b}}{{|\vec b|}}$ , this is called the projection formula.
4) If a vector $\vec v = a\hat i + b\hat j + c\hat k$ , then \[|\vec v| = \sqrt {{a^2} + {b^2} + {c^2}} \]

Complete step-by-step answer:
We are given two points passing through a line.
Let ‘a’ be the vector joining the points (2,0,-3) and (5,-1,2)
We will use the formula for the line equation in vector form.
Now we have $\vec a = (5 - 2)\hat i + ( - 1 - 0)\hat j + (2 + 3)\hat k$
$ \Rightarrow \vec a = 3\hat i - \hat j + 5\hat k$
We are also given the direction ratios of another line that is 2,4,4
Let it be ‘b’
Now, $\vec b = 2\hat i + 4\hat j + 4\hat k$
Thus by projection formula we have,
Projection of ‘a’ on ‘b’ is,
$\Rightarrow \dfrac{{\vec a \cdot \vec b}}{{|\vec b|}}$
$ = \dfrac{{(3\hat i - \hat j + 5\hat k) \cdot (2\hat i + 4\hat j + 4\hat k)}}{{|2\hat i + 4\hat j + 4\hat k|}}$
$ = \dfrac{{6 - 4 + 20}}{{\sqrt {4 + 16 + 16} }}$
$ = \dfrac{{22}}{{\sqrt {36} }} = \dfrac{{22}}{6} = \dfrac{{11}}{3}$
So, the correct answer is “Option e ”.

Note: Projection is also called shadow, you can also be asked the question for finding the shadow of a line passing through two points on a line whose direction ratios are given. These types of questions can be solved the same fashion as this one.