1.
If $l, m, n$ are the direction cosines of a line, write the fundamental identity connecting them.
2.
Write the direction ratios of the line joining the points $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$.
3.
If $a, b, c$ are the direction ratios of a line, write the formula to find its direction cosine '$l$'.
4.
What are the direction cosines of the x-axis?
5.
What are the direction cosines of the y-axis?
6.
What are the direction cosines of the z-axis?
7.
If a line makes angles $\alpha, \beta, \gamma$ with the coordinate axes, what is the value of $\cos^2\alpha + \cos^2\beta + \cos^2\gamma$?
8.
Using the result from the previous question, deduce the value of $\sin^2\alpha + \sin^2\beta + \sin^2\gamma$.
9.
Write the vector equation of a line passing through a point with position vector $\vec{a}$ and parallel to vector $\vec{b}$.
10.
Write the Cartesian equation of a line passing through $(x_1, y_1, z_1)$ with direction ratios $a, b, c$.
11.
Write the vector equation of a line passing through two points with position vectors $\vec{a}$ and $\vec{b}$.
12.
Write the Cartesian equation of a line passing through $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$.
13.
In the vector equation of a line $\vec{r} = \vec{a} + \lambda\vec{b}$, what does the scalar $\lambda$ represent?
14.
Write the coordinates of the general point on the line $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c} = \lambda$.
15.
Identify a point through which the line $\frac{x-2}{3} = \frac{y-4}{5} = \frac{z+1}{2}$ passes.
16.
Identify the direction ratios of the line $\frac{x-2}{3} = \frac{y-4}{5} = \frac{z+1}{2}$.
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 $\vec{r} = \vec{a_2} + \mu\vec{b_2}$.
18.
What is the condition for two lines to be perpendicular in terms of their direction vectors $\vec{b_1}$ and $\vec{b_2}$?
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.
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.
21.
What is the angle between the x-axis and the y-axis in 3D space?
22.
If $\vec{b_1} \cdot \vec{b_2} = 0$, what is the angle between the lines?
23.
Lines in space which are neither parallel nor intersecting are called _________________ lines.
24.
Write the vector formula for the shortest distance $d$ between two skew lines $\vec{r} = \vec{a_1} + \lambda\vec{b_1}$ and $\vec{r} = \vec{a_2} + \mu\vec{b_2}$.
25.
Write the vector formula for the shortest distance $d$ between two parallel lines $\vec{r} = \vec{a_1} + \lambda\vec{b}$ and $\vec{r} = \vec{a_2} + \mu\vec{b}$.
26.
If two lines intersect, what is the value of the shortest distance between them?
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 _________________.
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 $\lambda$.
29.
True/False: The direction vector of the perpendicular line and the direction vector of the given line have a dot product of zero.
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 $P'$?
31.
Write the general Cartesian equation of a plane.
32.
Write the normal form of the equation of a plane in vector form.
33.
In the equation $\vec{r} \cdot \hat{n} = d$, what does the scalar '$d$' physically represent?
34.
Write the equation of a plane passing through a point $(x_1, y_1, z_1)$ with a normal vector having direction ratios $A, B, C$.
35.
Write the intercept form of the equation of a plane having intercepts $a, b, c$ on the x, y, and z axes respectively.
36.
What is the equation of the $xy$-plane?
37.
What is the equation of the $yz$-plane?
38.
If a plane passes through the origin, what is the value of the constant term $D$ in its general equation $Ax+By+Cz+D=0$?
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 $\cos\theta =$ _________________.
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 $\sin\theta =$ _________________.
41.
What is the condition for a line (direction $\vec{b}$) to be parallel to a plane (normal $\vec{n}$)?
42.
What is the condition for a line (direction $\vec{b}$) to be perpendicular to a plane (normal $\vec{n}$)?
43.
Write the condition for two planes with normal vectors $\vec{n_1}$ and $\vec{n_2}$ to be mutually perpendicular.
44.
Write the condition for two planes with normal vectors $\vec{n_1}$ and $\vec{n_2}$ to be parallel.
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 _________________.
46.
Write the formula for the perpendicular distance of a point $(x_1, y_1, z_1)$ from the plane $Ax + By + Cz + D = 0$.
47.
Write the formula for the distance between two parallel planes $Ax + By + Cz + d_1 = 0$ and $Ax + By + Cz + d_2 = 0$.
48.
What is the perpendicular distance of the origin $(0,0,0)$ from the plane $Ax + By + Cz + D = 0$?
49.
Write the 3D distance formula to find the distance between point $(x_1, y_1, z_1)$ and the origin.
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.
