Question 1:
If a line makes angles of \(\frac{3\pi}{4}\), \(\frac{\pi}{3}\) and \(\theta\) with the positive directions of x, y and z-axis respectively, then \(\theta\) is
Question 2:
The equation of a line parallel to the vector \(3\hat{i}+\hat{j}+2\hat{k}\) and passing through the point \((4, -3, 7)\) is:
Question 3:
The line \(x=1+5\mu\), \(y=-5+\mu\), \(z=-6-3\mu\) passes through which of the following point ?
Question 4:
If P is a point on the line segment joining (3, 6, -1) and (6, 2, -2) and y-coordinate of P is 4, then its z-coordinate is:
Question 5:
If the direction cosines of a line are \(\sqrt{3}k, \sqrt{3}k\), \(\sqrt{3}k,\) then the value of k is:
Question 6:
The distance of point \(P(a,b,c)\) from y-axis is :
Question 7:
The coordinates of the foot of the perpendicular drawn from the point \((0, 1, 2)\) on the x-axis are given by:
Question 8:
Direction ratios of a vector parallel to line \(\frac{x-1}{2}=-y=\frac{2z+1}{6}\) are:
Question 9:
If a line makes an angle of \(30^{\circ}\) with the positive direction of x-axis, \(120^{\circ}\) with the positive direction of y-axis, then the angle which it makes with the positive direction of z-axis is:
Question 10:
If \(\alpha\), \(\beta\) and \(\gamma\) are the angles which a line makes with positive directions of x, y and z axes respectively, then which of the following is not true?
Question 11:
If a line makes an angle of \(\frac{\pi}{4}\) with the positive directions of both x-axis and z-axis, then the angle which it makes with the positive direction of y-axis is:
Question 12:
The vector equation of a line passing through the point (1, -1, 0) and parallel to Y-axis is :
Question 13:
The lines \(\frac{1-x}{2}=\frac{y-1}{3}=\frac{z}{1}\) and \(\frac{2x-3}{2p}=\frac{y}{-1}=\frac{z-4}{7}\) are perpendicular to each other for p equal to:
Question 14:
The angle which the line \(\frac{x}{1}=\frac{y}{-1}=\frac{z}{0}\) makes with the positive direction of Y-axis is:
Question 15:
The Cartesian equation of the line passing through the point (1, -3, 2) and parallel to the line: \(\vec{r}=(2+\lambda)\hat{i}+\lambda\hat{j}+(2\lambda-1)\hat{k}\) is:
Question 16:
Determine if the lines \(\vec{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda(3\hat{i}-\hat{j})\) and \(\vec{r}=(4\hat{i}-\hat{k})+\mu(2\hat{i}+3\hat{k})\) intersect with each other.
Question 17:
Find the vector equation of the line passing through the point (2, 3, -5) and making equal angles with the co-ordinate axes.
Question 18:
Find the distance of the point (-1, -5, -10) from the point of intersection of the lines \(\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}\) and \(\frac{x-4}{5}=\frac{y-1}{2}=z.\)
Question 19:
Find the distance of the point \(P(2,4,-1)\) from the line \(\frac{x+5}{1}=\frac{y+3}{4}=\frac{z-6}{-9}\).
Question 20:
Let the position vectors of the points A, B and C be \(3\hat{i}-\hat{j}-2\hat{k}\), \(\hat{i}+2\hat{j}-\hat{k}\) and \(\hat{i}+5\hat{j}+3\hat{k}\) respectively. Find the vector and cartesian equations of the line passing through A and parallel to line BC.
Question 21:
Find the shortest distance between the lines: \(\vec{r}=(2\hat{i}-\hat{j}+3\hat{k})+\lambda(\hat{i}-2\hat{j}+3\hat{k})\) and \(\vec{r}=(\hat{i}+4\hat{k})+\mu(3\hat{i}-6\hat{j}+9\hat{k}).\)
Question 22:
Find the image $A'$ of the point $A(1, 6, 3)$ in the line $\frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3}$. Also, find the equation of the line joining $A$ and $A'$.
Question 23:
Find a point $P$ on the line $\frac{x+5}{1} = \frac{y+3}{4} = \frac{z-6}{-9}$ such that its distance from point $Q(2, 4, -1)$ is 7 units. Also, find the equation of the line joining $P$ and $Q$.
Question 24:
Find the shortest distance between the lines: \(\frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}\) and \(\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5}\).
Question 25:
Find the image \(A'\) of the point \(A(2, 1, 2)\) in the line \(l:\vec{r}=4\hat{i}+2\hat{j}+2\hat{k}+\lambda(\hat{i}-\hat{j}-\hat{k})\). Also, find the equation of line joining \(AA'\). Find the foot of perpendicular from point A on the line \(l\).
Question 26:
Find the equation of a line in vector and cartesian form which passes through the point \((1,2,-4)\) and is perpendicular to the lines \(\frac{x-8}{3}=\frac{y+19}{-16}=\frac{z-10}{7}\) and \(\vec{r}=15\hat{i}+29\hat{j}+5\hat{k}+\mu(3\hat{i}+8\hat{j}-5\hat{k}).\)
Question 27:
Find the foot of the perpendicular drawn from the point \((1, 1, 4)\) on the line \(\frac{x+2}{5}=\frac{y+1}{2}=\frac{-z+4}{-3}\).
Question 28:
Find the point on the line \(\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-4}{3}\) at a distance of \(2\sqrt{2}\) units from the point \((-1, -1, 2)\).
Question 29:
Let the polished side of the mirror be along the line \(\frac{x}{1}=\frac{1-y}{-2}=\frac{2z-4}{6}.\) A point \(P(1,6,3)\), some distance away from the mirror, has its image formed behind the mirror. Find the coordinates of the image point and the distance between the point P and its image.
Question 30:
Find the equation of the plane passing through the points \((3, 1, 1)\), \((0, 2, 4)\) and \((-2, 1, -1)\).
Question 31:
Find the point Q on the line \(\frac{2x+4}{6}=\frac{y+1}{2}=\frac{-2z+6}{-4}\) at a distance of \(3\sqrt{2}\) from the point \(P(1,2,3)\).
Question 32:
Find the image of the point \((-1,5,2)\) in the line \(\frac{2x-4}{2}=\frac{y}{2}=\frac{2-z}{3}\). Find the length of the line segment joining the points (given point and the image point).
Question 33:
Find the co-ordinates of the foot of the perpendicular drawn from the point (2, 3, -8) to the line \(\frac{4-x}{2}=\frac{y}{6}=\frac{1-z}{3}\).Also, find the perpendicular distance of the given point from the line.
Question 34:
Find the distance between the line \(\frac{x}{2}=\frac{2y-6}{4}=\frac{1-z}{-1}\) and another line parallel to it passing through the point (4, 0, -5).
Question 35:
Find the equation of the line passing through the point of intersection of the lines \(\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}\) and \(\frac{x-1}{0}=\frac{y}{-3}=\frac{z-7}{2}\) and perpendicular to these given lines.
Question 36:
Find the shortest distance between the lines \(L_{1}\) & \(L_{2}\) given below: \(L_{1}\): The line passing through (2, -1, 1) and parallel to \(\frac{x}{1}=\frac{y}{1}=\frac{z}{3}\).\(L_{2}:\vec{r}=\hat{i}+(2\mu+1)\hat{j}-(\mu+2)\hat{k}\).
Question 37:
If the lines \(\frac{x-1}{-3}=\frac{y-2}{2k}=\frac{z-3}{2}\) and \(\frac{x-1}{3k}=\frac{y-1}{1}=\frac{z-6}{-7}\) are perpendicular to each other, find the value of k and hence write the vector equation of a line perpendicular to these two lines and passing through the point (3, -4, 7).
Question 38:
Two vertices of the parallelogram ABCD are given as \(A(-1,2,1)\) and \(B(1,-2,5)\). If the equation of the line passing through C and D is \(\frac{x-4}{1}=\frac{y+7}{-2}=\frac{z-8}{2}\), then find the distance between sides AB and CD. Hence, find the area of parallelogram ABCD.
Question 39:
Find the equation of the line which bisects the line segment joining points \(A(2,3,4)\) and \(B(4,5,8)\) and is perpendicular to the lines \(\frac{x-8}{3}=\frac{y+19}{-16}=\frac{z-10}{7}\) and \(\frac{x-15}{3}=\frac{y-29}{8}=\frac{z-5}{-5}\)
Question 40:
The image of point \(P(x,y,z)\) with respect to line \(\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}\) is \(P^{\prime}(1,0,7)\). Find the coordinates of point P.