1.
If $l, m, n$ are the direction cosines of a line, write the fundamental identity connecting them.
Answer: $l^2 + m^2 + n^2 = 1$
2.
Write the direction ratios of the line joining the points $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$.
Answer: $(x_2 - x_1), (y_2 - y_1), (z_2 - z_1)$
3.
If $a, b, c$ are the direction ratios of a line, write the formula to find its direction cosine '$l$'.
Answer: $l = \pm \frac{a}{\sqrt{a^2 + b^2 + c^2}}$
4.
What are the direction cosines of the x-axis?
Answer: $1, 0, 0$
5.
What are the direction cosines of the y-axis?
Answer: $0, 1, 0$
6.
What are the direction cosines of the z-axis?
Answer: $0, 0, 1$
7.
If a line makes angles $\alpha, \beta, \gamma$ with the coordinate axes, what is the value of $\cos^2\alpha + \cos^2\beta + \cos^2\gamma$?
Answer: $1$
8.
Using the result from the previous question, deduce the value of $\sin^2\alpha + \sin^2\beta + \sin^2\gamma$.
Answer: $(1-\cos^2\alpha) + (1-\cos^2\beta) + (1-\cos^2\gamma) = 3 - 1 = 2$
9.
Write the vector equation of a line passing through a point with position vector $\vec{a}$ and parallel to vector $\vec{b}$.
Answer: $\vec{r} = \vec{a} + \lambda\vec{b}$
10.
Write the Cartesian equation of a line passing through $(x_1, y_1, z_1)$ with direction ratios $a, b, c$.
Answer: $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$
11.
Write the vector equation of a line passing through two points with position vectors $\vec{a}$ and $\vec{b}$.
Answer: $\vec{r} = \vec{a} + \lambda(\vec{b} - \vec{a})$
12.
Write the Cartesian equation of a line passing through $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$.
Answer: $\frac{x-x_1}{x_2-x_1} = \frac{y-y_1}{y_2-y_1} = \frac{z-z_1}{z_2-z_1}$
13.
In the vector equation of a line $\vec{r} = \vec{a} + \lambda\vec{b}$, what does the scalar $\lambda$ represent?
Answer: A real parameter (or constant) corresponding to different points on the line.
14.
Write the coordinates of the general point on the line $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c} = \lambda$.
Answer: $(a\lambda + x_1, b\lambda + y_1, c\lambda + z_1)$
15.
Identify a point through which the line $\frac{x-2}{3} = \frac{y-4}{5} = \frac{z+1}{2}$ passes.
Answer: $(2, 4, -1)$
16.
Identify the direction ratios of the line $\frac{x-2}{3} = \frac{y-4}{5} = \frac{z+1}{2}$.
Answer: $3, 5, 2$
17.
Write the formula for $\cos\theta$, where $\theta$ is the angle between the lines $\vec{r} = \vec{a_1} + \lambda\vec{b_1}$ and $\vec{r} = \vec{a_2} + \mu\vec{b_2}$.
Answer: $\cos\theta = \left|\frac{\vec{b_1} \cdot \vec{b_2}}{|\vec{b_1}||\vec{b_2}|}\right|$
18.
What is the condition for two lines to be perpendicular in terms of their direction vectors $\vec{b_1}$ and $\vec{b_2}$?
Answer: $\vec{b_1} \cdot \vec{b_2} = 0$
19.
If two lines have direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$, write the Cartesian condition for them to be perpendicular.
Answer: $a_1 a_2 + b_1 b_2 + c_1 c_2 = 0$
20.
If two lines have direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$, write the Cartesian condition for them to be parallel.
Answer: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$
21.
What is the angle between the x-axis and the y-axis in 3D space?
Answer: $90^\circ$ (or $\frac{\pi}{2}$)
22.
If $\vec{b_1} \cdot \vec{b_2} = 0$, what is the angle between the lines?
Answer: $90^\circ$ (The lines are perpendicular)
23.
Lines in space which are neither parallel nor intersecting are called _________________ lines.
Answer: Skew
24.
Write the vector formula for the shortest distance $d$ between two skew lines $\vec{r} = \vec{a_1} + \lambda\vec{b_1}$ and $\vec{r} = \vec{a_2} + \mu\vec{b_2}$.
Answer: $d = \left| \frac{(\vec{b_1} \times \vec{b_2}) \cdot (\vec{a_2} - \vec{a_1})}{|\vec{b_1} \times \vec{b_2}|} \right|$
25.
Write the vector formula for the shortest distance $d$ between two parallel lines $\vec{r} = \vec{a_1} + \lambda\vec{b}$ and $\vec{r} = \vec{a_2} + \mu\vec{b}$.
Answer: $d = \left| \frac{\vec{b} \times (\vec{a_2} - \vec{a_1})}{|\vec{b}|} \right|$
26.
If two lines intersect, what is the value of the shortest distance between them?
Answer: $0$ (Zero)
27.
To prove that two skew lines intersect, the scalar triple product $(\vec{b_1} \times \vec{b_2}) \cdot (\vec{a_2} - \vec{a_1})$ must be equal to _________________.
Answer: $0$
28.
To find the foot of the perpendicular from point $P$ to a line, we first assume the foot to be the _________________ point on the line in terms of $\lambda$.
Answer: General
29.
True/False: The direction vector of the perpendicular line and the direction vector of the given line have a dot product of zero.
Answer: True (Since they are perpendicular).
30.
If $F$ is the foot of the perpendicular from $P$ to a line, and $P'$ is the image of $P$, what is the geometric relationship between $P, F,$ and $P'$?
Answer: $F$ is the midpoint of the line segment joining $P$ and $P'$.
31.
Write the general Cartesian equation of a plane.
Answer: $Ax + By + Cz + D = 0$
32.
Write the normal form of the equation of a plane in vector form.
Answer: $\vec{r} \cdot \hat{n} = d$
33.
In the equation $\vec{r} \cdot \hat{n} = d$, what does the scalar '$d$' physically represent?
Answer: The perpendicular distance of the plane from the origin.
34.
Write the equation of a plane passing through a point $(x_1, y_1, z_1)$ with a normal vector having direction ratios $A, B, C$.
Answer: $A(x - x_1) + B(y - y_1) + C(z - z_1) = 0$
35.
Write the intercept form of the equation of a plane having intercepts $a, b, c$ on the x, y, and z axes respectively.
Answer: $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$
36.
What is the equation of the $xy$-plane?
Answer: $z = 0$
37.
What is the equation of the $yz$-plane?
Answer: $x = 0$
38.
If a plane passes through the origin, what is the value of the constant term $D$ in its general equation $Ax+By+Cz+D=0$?
Answer: $D = 0$
39.
The angle $\theta$ between two planes $\vec{r} \cdot \vec{n_1} = d_1$ and $\vec{r} \cdot \vec{n_2} = d_2$ is given by $\cos\theta =$ _________________.
Answer: $\left|\frac{\vec{n_1} \cdot \vec{n_2}}{|\vec{n_1}||\vec{n_2}|}\right|$
40.
The angle $\theta$ between a line $\vec{r} = \vec{a} + \lambda\vec{b}$ and a plane $\vec{r} \cdot \vec{n} = d$ is given by $\sin\theta =$ _________________.
Answer: $\left|\frac{\vec{b} \cdot \vec{n}}{|\vec{b}||\vec{n}|}\right|$
41.
What is the condition for a line (direction $\vec{b}$) to be parallel to a plane (normal $\vec{n}$)?
Answer: $\vec{b} \cdot \vec{n} = 0$
42.
What is the condition for a line (direction $\vec{b}$) to be perpendicular to a plane (normal $\vec{n}$)?
Answer: $\vec{b}$ is parallel to $\vec{n}$ ($\vec{b} = k\vec{n}$)
43.
Write the condition for two planes with normal vectors $\vec{n_1}$ and $\vec{n_2}$ to be mutually perpendicular.
Answer: $\vec{n_1} \cdot \vec{n_2} = 0$
44.
Write the condition for two planes with normal vectors $\vec{n_1}$ and $\vec{n_2}$ to be parallel.
Answer: $\vec{n_1} = \lambda\vec{n_2}$ (Their normals are proportional)
45.
Two lines $\vec{r} = \vec{a_1} + \lambda\vec{b_1}$ and $\vec{r} = \vec{a_2} + \mu\vec{b_2}$ are coplanar if their shortest distance is _________________.
Answer: $0$
46.
Write the formula for the perpendicular distance of a point $(x_1, y_1, z_1)$ from the plane $Ax + By + Cz + D = 0$.
Answer: $\frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}}$
47.
Write the formula for the distance between two parallel planes $Ax + By + Cz + d_1 = 0$ and $Ax + By + Cz + d_2 = 0$.
Answer: $\frac{|d_1 - d_2|}{\sqrt{A^2 + B^2 + C^2}}$
48.
What is the perpendicular distance of the origin $(0,0,0)$ from the plane $Ax + By + Cz + D = 0$?
Answer: $\frac{|D|}{\sqrt{A^2 + B^2 + C^2}}$
49.
Write the 3D distance formula to find the distance between point $(x_1, y_1, z_1)$ and the origin.
Answer: $\sqrt{x_1^2 + y_1^2 + z_1^2}$
50.
If a plane has the intercept form $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$, write the coordinates of its intersection with the z-axis.
Answer: $(0, 0, c)$
