Mathematics Multiple Choice Questions on “Three Dimensional Geometry – Angle between Two Lines”.
1. If L1 and L2 have the direction ratios (a_1,b_1,c_1 ,and ,a_2,b_2,c_2) respectively then what is the angle between the lines?
a) (θ=tan^{-1}left|frac{a_1 a_2+b_1 b_2+c_1 c_2}{sqrt{a_1^2+b_1^2+c_1^2} sqrt{a_2^2+b_2^2+c_2^2}}right |)
b) (θ=2tan^{-1}left|frac{a_1 a_2+b_1 b_2+c_1 c_2}{sqrt{a_1^2+b_1^2+c_1^2} sqrt{a_2^2+b_2^2+c_2^2}}right |)
c) (θ=cos^{-1}left|frac{a_1 a_2+b_1 b_2+c_1 c_2}{sqrt{a_1^2+b_1^2+c_1^2} sqrt{a_2^2+b_2^2+c_2^2}}right |)
d) (θ=2 ,cos^{-1}left|frac{a_1 a_2+b_1 b_2+c_1 c_2}{sqrt{a_1^2+b_1^2+c_1^2} sqrt{a_2^2+b_2^2+c_2^2}}right |)
Answer: c
Clarification: If L1 and L2 have the direction ratios (a_1,b_1,c_1 ,and ,a_2,b_2,c_2) respectively then the angle between the lines is given by
(cosθ=left|frac{a_1 a_2+b_1 b_2+c_1 c_2}{sqrt{a_1^2+b_1^2+c_1^2} sqrt{a_2^2+b_2^2+c_2^2}}right |)
(θ=cos^{-1}left|frac{a_1 a_2+b_1 b_2+c_1 c_2}{sqrt{a_1^2+b_1^2+c_1^2} sqrt{a_2^2+b_2^2+c_2^2}}right |)
2. Find the angle between the lines.
(frac{x+2}{1}=frac{y+5}{6}=frac{z-3}{2})
(frac{x-4}{5}=frac{y-3}{-2}=frac{z+3}{1})
a) (cos^{-1}frac{5}{sqrt{1230}})
b) (cos^{-1}frac{3}{sqrt{3120}})
c) (cos^{-1}frac{7}{sqrt{2310}})
d) (cos^{-1}frac{48}{sqrt{1230}})
Answer: a
Clarification: We know that, the angle between two lines is given by the formula
cosθ=(left |frac{a_1 a_2+b_1 b_2+c_1 c_2}{sqrt{a_1^2+b_1^2+c_1^2} sqrt{a_2^2+b_2^2+c_2^2}}right |)
cosθ=(left |frac{1(5)+6(-2)+2(1)}{sqrt{1^2+6^2+2^2).√(5^2+(-2)^2+1^2}}right |)
=(left |frac{-5}{sqrt{41}.sqrt{30}} right |=frac{5}{sqrt{1230}})
∴(θ=cos^{-1}frac{5}{sqrt{1230}})
3. Find the value of p such that the lines
(frac{x-1}{3}=frac{y+4}{p}=frac{z-9}{1})
(frac{x+2}{1}=frac{y-3}{1}=frac{z-7}{-2})
are at right angles to each other.
a) p=2
b) p=1
c) p=-1
d) p=-2
Answer: c
Clarification: The angle between two lines is given by the equation
(cosθ=left |frac{a_1 a_2+b_1 b_2+c_1 c_2}{sqrt{a_1^2+b_1^2+c_1^2} sqrt{a_2^2+b_2^2+c_2^2}}right |)
cos90°=(left |frac{3(1)+p(1)+1(-2)}{sqrt{3^2+p^2+1^2}.sqrt{1^2+1^2+(-2)^2}}right |)
0=(|frac{p+1}{sqrt{10+p^2}.√6}|)
0=p+1
p=-1
4. Find the angle between the two lines if the equations of the lines are
(vec{r}=hat{i}+hat{j}+hat{k}+λ(3hat{i}-hat{j}+hat{k}) ,and ,vec{r}=4hat{i}+hat{j}-2hat{k}+μ(2hat{i}+3hat{j}+hat{k}))
a) (cos^{-1}frac{4}{sqrt{14}})
b) (cos^{-1}frac{7}{sqrt{154}})
c) (cos^{-1}frac{4}{154})
d) (cos^{-1}frac{4}{sqrt{154}})
Answer: d
Clarification: Given that, (vec{r}=hat{i}+hat{j}+hat{k}+λ(3hat{i}-hat{j}+hat{k})) and (vec{r}=4hat{i}+hat{j}-2hat{k}+μ(2hat{i}+3hat{j}+hat{k}))
We know that, if the equations of two lines are of the form (vec{r}=vec{a_1}+λvec{b_1} and ,vec{r}=vec{a_2}+μvec{b_2}) then the angle between the two lines is given by
(cosθ=left|frac{vec{b_1}.vec{b_2}}{|vec{b_1}||vec{b_2}|}right|)
=(left |frac{(3(2)+(-1)3+1(1)}{sqrt{3^2+(-1)^2+(1)^2.} sqrt{2^2+3^2+1^2}}right |=frac{4}{sqrt{11}.sqrt{14}}=frac{4}{sqrt{154}})
θ=(cos^{-1}frac{4}{sqrt{154}}).
5. If two lines L1 and L2 with direction ratios (a_1,b_1,c_1 ,and ,a_2,b_2,c_2) respectively are perpendicular to each other then
(a_1 a_2+b_1 b_2+c_1 c_2=0)
a) True
b) False
Answer: a
Clarification: The given statement is true.
We know that the angle between two lines is given by the formula
cosθ=(left |frac{a_1 a_2+b_1 b_2+c_1 c_2}{sqrt{a_1^2+b_1^2+c_1^2} sqrt{a_2^2+b_2^2+c_2^2}}right |)
So, if the lines L1 and L2 are perpendicular to each to each other then,
θ=90°
⟹(a_1 ,a_2+b_1 ,b_2+c_1 ,c_2)=0
6. Find the value of p such that the lines (frac{x+11}{4}=frac{y+3}{-2}=frac{z-3}{4} ,and ,frac{x-3}{p}=frac{y+12}{2}=frac{z-3}{-12}) are at right angles to each other.
a) p=11
b) p=12
c) p=13
d) p=4
Answer: c
Clarification: We know that, if two lines are perpendicular to each other then,
(a_1 a_2+b_1 b_2+c_1 c_2=0)
i.e.4(p)+(-2)2+4(-12)=0
4p-4-48=0
4p=52
p=(frac{52}{4})=13.
7. If the equations of two lines L1 and L2 are (vec{r}=vec{a_1}+λvec{b_1}) and (vec{r}=vec{a_2}+μvec{b_2}), then which of the following is the correct formula for the angle between the two lines?
a) cosθ=(left |frac{vec{a_1}.vec{a_2}}{|vec{b_1}||vec{a_2}|}right |)
b) cosθ=(left |frac{vec{a_1}.vec{a_2}}{|vec{a_1}||vec{a_2}|}right |)
c) cosθ=(left |frac{vec{b_1}.vec{b_2}}{|vec{b_1}||vec{b_2}|}right |)
d) cosθ=(left |frac{vec{a_1}.vec{b_2}}{|vec{a_1}||vec{b_2}|}right |)
Answer: c
Clarification: Given that the equations of the lines are
(vec{r}=vec{a_1}+λvec{b_1} ,and ,vec{r}=vec{a_2}+μvec{b_2})
∴ the angle between the two lines is given by
cosθ=(left |frac{vec{b_1}.vec{b_2}}{|vec{b_1}||vec{b_2}|}right |).
8. Find the angle between the lines (vec{r}=2hat{i}+6hat{j}-hat{k}+λ(hat{i}-2hat{j}+3hat{k})) and (vec{r}=4hat{i}-7hat{j}+3hat{k}+μ(5hat{i}-3hat{j}+3hat{k})).
a) θ=(cos^{-1}frac{20}{sqrt{602}})
b) θ=(cos^{-1}frac{20}{sqrt{682}})
c) θ=(cos^{-1}frac{8}{sqrt{602}})
d) θ=(cos^{-1}frac{14}{sqrt{598}})
Answer: a
Clarification: If two lines have the equations (vec{r}=vec{a_1}+λvec{b_1} ,and ,vec{r}=vec{a_2}+μvec{b_2})
Then, the angle between the two lines will be given by
cosθ=(left |frac{vec{b_1}.vec{b_2}}{|vec{b_1}||vec{b_2}|}right |)
=(left |frac{(hat{i}-2hat{j}+3hat{k}).(5hat{i}-3hat{j}+3hat{k})}{sqrt{1^2+(-2)^2+(3)^2).√(5^2+(-3)^2+3^2}}right |)
=(frac{5+6+9}{√14.√43}=frac{20}{√602})
θ=(cos^{-1}frac{20}{sqrt{602}})
9. If two lines L1 and L2 are having direction cosines (l_1,m_1,n_1 ,and ,l_2,m_2,n_2) respectively, then what is the angle between the two lines?
a) cotθ=(left |l_1 ,l_2+m_1 ,m_2+n_1 ,n_2right |)
b) sinθ=(left |l_1 ,l_2+m_1 ,n_2+n_1 ,m_2right |)
c) tanθ=(left |l_1 ,l_2+m_1 ,m_2+n_1 ,n_2right |)
d) cosθ=(left |l_1 ,l_2+m_1 ,m_2+n_1 ,n_2right |)
Answer: d
Clarification: If two lines L1 and L2 are having direction cosines (l_1,m_1,n_1 ,and ,l_2,m_2,n_2) respectively, then the angle between the lines is given by
cosθ=(left |l_1 ,l_2+m_1 ,m_2+n_1 ,n_2right |)
10. Find the angle between the pair of lines (frac{x-3}{5}=frac{y+7}{3}=frac{z-2}{2} ,and ,frac{x+1}{3}=frac{y-5}{4}=frac{z+2}{8}).
a) (cos^{-1}frac{43}{sqrt{3482}})
b) (cos^{-1}frac{43}{sqrt{3382}})
c) (cos^{-1}frac{85}{sqrt{3382}})
d) (cos^{-1}frac{34}{sqrt{3382}})
Answer: b
Clarification: The direction ratios are 5, 3, 2 for L1 and 3, 4, 8 for L2
∴ the angle between the two lines is given by
cosθ=(frac{(a_1 a_2+b_1 b_2+c_1 c_2)}{sqrt{a_1^2+b_1^2+c_1^2} sqrt{a_2^2+b_2^2+c_2^2}})
=(frac{15+12+16}{sqrt{5^2+3^2+2^2}.sqrt{3^2+4^2+8^2}})
=(frac{43}{sqrt{38}.sqrt{89}}=frac{43}{sqrt{3382}})
θ=(cos^{-1}frac{43}{sqrt{3382}}).