Direction Ratios

Q. In $R^3$, Let $L$ be a straight line passing through the origin. Suppose that all the points on $L$ are at a constant distance from the two planes $P_1:x+2y-z+1=0$ and $P_2:2x-y+z-1=0.$ Let $M$ be the locus of the feet of the perpendiculars drawn from the points on $L$ on the plane $P_1.$ Which of the following points lie(s) on $M?$

1. $(0,-\frac{5}{6},-\frac{2}{3})$
2. $(-\frac{1}{6},-\frac{1}{3},\frac{1}{6})$
3. $(-\frac{5}{6},0,\frac{1}{6})$
4. $(-\frac{1}{3},0,\frac{2}{3})$

Q. Let $P_1:\overrightarrow{r}.(2\hat{i}+\hat{j}-3\hat{k})=4$ be a plane. Let $P_2$ be another plane which passes through the points $(2,–3,2),(2,–2,–3)$ and $(1,–4,2).$ If the direction ratios of the line of intersection of $P_1$ and $P_2$ be $16,$ $\alpha ,\beta ,$ then the value of $\alpha +\beta$ is equal to

Q. Let a line having direction ratios $1,-4,2$ intersect the lines $\frac{x-7}{3}=\frac{y-1}{-1}=\frac{z+2}{1}$ and $\frac{x}{2}=\frac{y-7}{3}=\frac{z}{1}$ at the points $A$ and $B$. Then $(AB{)}^2$ is equal to

Q. $ABC$ is a triangle in a plane with vertices $A(2,3,5),B(-1,3,2)$ and $C(\lambda ,5,\mu ).$ If the median through $A$ is equally inclined to the coordinate axes, then the value of $({\lambda }^3+{\mu }^3+5)$ is:

1. $1130$
2. $1077$
3. $676$
4. $1348$

Q. The plane $2x-2y+z=3$ is rotated about the line where it cuts the $xy-$plane by an acute angle $\alpha$. If the new position of plane contains the point $(3,1,1)$ then $9\cos \alpha$ is equal to

Q.

Two lines whose direction ratios are a₁, b₁, c₁ and a₂, b₂, c₂ are perpendicular, if

1. 

2. 

3. 

4. 

Q. Find the direction ratios of a line perpendicular to the plane 2x+5y-6z = 7

1. (2, 5, 6)
2. (-2, 5, 6)
3. (2, -5, 6)
4. (2, 5, -6)

Q. The plane denoted by $P_1:4x+7y+4z+81=0$ is rotated through a right angle about its line of intersection with the plane $P_2:5x+3y+10z=25$. If the plane in its new position is denoted by $P$, and the distance of this plane from the origin is $k$, then the value of $\left[\frac{k}{2}\right]$ is (where [.] represents greatest integer less than or equal to $k$).

Q. If $l_1,m_1,n_1$ and $l_2,m_2,n_2$ are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are $(m_1{n}_2-m_2{n}_1),(n_1{l}_2-n_2{l}_1),$

Q. If the direction ratios of a line are (1, 1, 2), then find the direction cosines of the line.

1. (1, 1, 2)
2. $\left(0,\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}}\right)$
3. $\left(\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}}\right)$
4. $\left(\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}},\frac{2}{\sqrt{6}}\right)$

Q. A physical quantity A is related to four observable a, b, c and d as follows, $A=\frac{a^2{b}^3}{c\sqrt{d}}$, the percentage errors of measurement in a, b, c and d are $1\%,3\%,2\%$ and $2\%$ respectively. The maximum error in the quantity A is

1. $7\%$
2. $5\%$
3. $12\%$
4. $14\%$

Q. If the direction cosines of a line are l, m and n and the direction ratios of the same line are a, b & c then which of the following is always correct ?

1. $\frac{l}{a}=\frac{m}{b}=\frac{n}{c}=1$
2. $\frac{l}{a}=\frac{m}{b}=\frac{n}{c}=2$
3. $\frac{l}{a}=\frac{m}{b}=\frac{n}{c}=K$, where $‘K^{\prime }$ is any constant real number.
4. None of these

Q. If 1, 2, 1 are the direction ratios of a line then which of the following could be direction cosines of this line?

1. 
2. 
3. 
4. None of these

Q.

If a line has the direction ratios −18, 12, −4, then what are its direction cosines?

Q. The distance of the point $(2,1,3)$ from the line $\frac{x-5}{4}=\frac{y+1}{3}=\frac{z-2}{6}$ is

1. $\sqrt{14}$
2. $\sqrt{26}$
3. $\sqrt{21}$
4. $7$

Q. The plane denoted by $P:4x+7y+4z+81=0$ is rotated through a right angle about its line of intersection with the plane $P_2:5x+3y+10z=25.$ If the plane in its new position be denoted by $P_1,$ and the distance of this plane from the origin is $d$, then the value of $\left[\frac{d}{2}\right]$ is $($ where $\left[k\right],k\in R$ represents the greatest integer less than or equal to $k)$

Q. Direction ratios of the normal to the plane passing through the points (0, 1, 1), (1, 1, 2)and (-1, 2, - 2) are

1. (1, 1, 1)
2. (2, 1, -1)
3. (1, 2, -1)
4. (1, -2, -1)

Q.

If the plane 2ax - 3ay + 4az + 6 = 0 passes through the midpoint of the line joining the centres of the spheres and $\begin{array}{c}{x}^2+y^2+z^2+6x-8y-2z=13\\ x^2+y^2+z^2-10x+4y-2z=8\end{array}$, then a equals
[AIEEE 2005]

1. -2

2. 2

3. -1

4. 1

Q. Using integration, find the area of the region in the first quadrant enclosed by the $x-axis$, the line $y=x$ and the circle $x^2+y^2=32$.

Q. The angle between the lines whose direction cosines satisfy the equations $l+m+n=0,l^2+m^2-n^2=0$, is given by

1. $\frac{\pi }{6}$
2. $\frac{5\pi }{3}$
3. $\frac{2\pi }{3}$
4. $\frac{\pi }{3}$

Q. If 1, 2, 1 are the direction ratios of a line then the direction cosines of this line is equal to

1. $\frac{2}{\sqrt{6}},\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}}$
2. $\frac{1}{\sqrt{6}},\frac{2}{\sqrt{6}},\frac{1}{\sqrt{6}}$
3. $\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}},\frac{2}{\sqrt{6}}$
4. $\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}}$

Q. In $R^3$, Let $L$ be a straight line passing through the origin. Suppose that all the points on $L$ are at a constant distance from the two planes $P_1:x+2y-z+1=0$ and $P_2:2x-y+z-1=0.$ Let $M$ be the locus of the feet of the perpendiculars drawn from the points on $L$ on the plane $P_1.$ Which of the following points lie(s) on $M?$

1. $(0,-\frac{5}{6},-\frac{2}{3})$
2. $(-\frac{1}{6},-\frac{1}{3},\frac{1}{6})$
3. $(-\frac{5}{6},0,\frac{1}{6})$
4. $(-\frac{1}{3},0,\frac{2}{3})$

Q. The direction ratios of a normal to the plane $OAB$ where $O=(0,0,0),A=(-1,2,3)$ and $B=(-3,4,5)$ are

1. $1,2,-1$
2. $2,1,-1$
3. $1,1,2$
4. $1,-1,2$

Q. The direction cosines of a line are l, m and n, and the direction ratios of the same line are a, b & c. Then, the expression is always correct.

1. $la+mb+nc=1$
2. $\frac{l}{a}=\frac{m}{b}=\frac{n}{c}=k$
3. $\frac{l}{a}=\frac{m}{b}=\frac{n}{c}=1$

Q.

If a line has the direction ratios -18, 12, -4, then what are its direction cosines?

Q. Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.

Q. The plane denoted by $P_1:4x+7y+4z+81=0$ is rotated through a right angle about its line of intersection with the plane $P_2:5x+3y+10z=25$. If the plane in its new position is denoted by $P$, and the distance of this plane from the origin is $k$, then the value of $\left[\frac{k}{2}\right]$ is (where [.] represents greatest integer less than or equal to $k$).

Q. Coordinates of a point $P$ lying on the line $\frac{x-2}{4}=\frac{y-1}{3}=\frac{z+1}{2}$ such that the line joining the points $P$ and $A(3,1,5)$ is parallel to the plane $4x-2y+z=0$, is

1. $(2,1,-1)$
2. $(\frac{16}{3},\frac{7}{2},\frac{2}{3})$
3. $(4,\frac{5}{2},0)$
4. $(3,\frac{7}{4},\frac{-1}{2})$

Q. If the direction cosines of a line are l, m and n and the direction ratios of the same line are a, b & c then which of the following is always correct ?

1. None of these
2. $\frac{l}{a}=\frac{m}{b}=\frac{n}{c}=1$
3. $\frac{l}{a}=\frac{m}{b}=\frac{n}{c}=2$
4. $\frac{l}{a}=\frac{m}{b}=\frac{n}{c}=K$, where $‘K^{\prime }$ is any constant real number.

Q. If a line has the direction ratios $-18,12,-4$, then what are its direction cosines?