Question

Let $l_{1}, m_{1}, n_{1}$ and $l_{2}, m_{2}, n_{2}$ be the direction cosines of two st-lines. Both the lines are perpendicular to each other, if-

$l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} = 0$
$l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} = 1$
$\frac{l_{1}}{l_{2}} = \frac{m_{1}}{m_{2}} = \frac{n_{1}}{n_{2}}$
$\frac{l_{1}}{l_{2}} + \frac{m_{1}}{m_{2}} + \frac{n_{1}}{n_{2}} = 0$

Answer: $l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} = 0$

Solution :-

Let $L_{1} and L_{2}$ are 1st and 2nd st-lines respectively.
Given that direction cosines of these two lines $L_{1} and L_{2}$ are $(l_{1}, m_{1}, n_{1}) and (l_{2}, m_{2}, n_{2})$ respectively. And, also given that $L_{1} ⊥ L_{2}$ .

Now, let’s consider $L_{1} and L_{2}$ as vectors. So, vector $L_{1} and L_{2}$ are given in component form $(l_{1}, m_{1}, n_{1}) and (l_{2}, m_{2}, n_{2})$ respectively.

If two vectors are given in component form lets say $(l_{1}, m_{1}, n_{1}) and (l_{2}, m_{2}, n_{2})$ then dot product of those two vectors are $l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2}$ .

Now, also we know that if two vectors are perpendicular to each other then dot product of those two vectors are zero. Because of angle between those two vectors are zero.

Hence, $L_{1} \cdot L_{2} = 0$

l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} = 0 \cdot

So, correct answer is option-1, $\to l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} = 0$

#Three Dimensional Geometry #BSEB PYQ 2017 #Mathematics #BSEB PYQs

Question

Let l1,m1,n1 and l2,m2,n2 be the direction cosines of two st-lines. Both the lines are perpendicular to each other, if-

Solution :-

Let $l_{1}, m_{1}, n_{1}$ and $l_{2}, m_{2}, n_{2}$ be the direction cosines of two st-lines. Both the lines are perpendicular to each other, if-