Find the position vector of a point A in space such that is inclined at 60° to and at 45° to and = 10 units.
Find the vector equation of the line which is parallel to the vector and which passes through the point (1,–2,3).
Show that the lines
and intersect.
Also, find their point of intersection.
Find the angle between the lines
and
Prove that the line through A (0, –1, –1) and B (4, 5, 1) intersects the line through C (3, 9, 4) and D (– 4, 4, 4).
Prove that the lines x = py + q, z = ry + s and x = p′y + q′, z = r′y + s′ are perpendicular if pp′ + rr′ + 1 = 0.
Find the equation of a plane which bisects perpendicularly the line joining the points A (2, 3, 4) and B (4, 5, 8) at right angles.
Find the equation of a plane which is at a distance units from origin and the normal to which is equally inclined to coordinate axis.
If the line drawn from the point (–2, – 1, – 3) meets a plane at right angle at the point (1, – 3, 3), find the equation of the plane.
Find the equation of the plane through the points (2, 1, 0), (3, –2, –2) and (3, 1, 7).
Find the equations of the two lines through the origin which intersect the line at angles of π/3 each.
Find the angle between the lines whose direction cosines are given by the equations l + m + n = 0, l2 + m2 – n2 = 0.
If a variable line in two adjacent positions has direction cosines l, m, n and l + δl, m + δm, n + δn, show that the small angle δθ between the two positions is given by
δθ2 = δl2 + δm2 + δn2
O is the origin and A is (a, b, c). Find the direction cosines of the line OA and the equation of plane through A at right angle to OA.
Two systems of rectangular axis have the same origin. If a plane cuts them at distances a, b, c and a′, b′, c′, respectively, from the origin, prove that
.
Find the foot of perpendicular from the point (2,3,–8) to the line . Also, find the perpendicular distance from the given point to the line.
Find the distance of a point (2,4,–1) from the line
Find the length and the foot of perpendicular from the point to the plane 2x – 2y + 4z + 5 = 0.
Find the equations of the line passing through the point (3,0,1) and parallel to the planes x + 2y = 0 and 3y – z = 0.
Find the equation of the plane through the points (2,1,–1) and (–1,3,4), and perpendicular to the plane x – 2y + 4z = 10.
Find the shortest distance between the lines given by and .
Find the equation of the plane which is perpendicular to the plane 5x + 3y + 6z + 8 = 0 and which contains the line of intersection of the planes x + 2y + 3z – 4 = 0 and 2x + y – z + 5 = 0.
The plane ax + by = 0 is rotated about its line of intersection with the plane z = 0 through an angle α. Prove that the equation of the plane in its new position is z = 0.
Find the equation of the plane through the intersection of the planes and whose perpendicular distance from origin is unity.
Show that the points and are equidistant from the plane and lies on opposite side of it.
and are two vectors. The position vectors of the points A and C are and , respectively. Find the position vector of a point P on the line AB and a point Q on the line CD such that PQ is perpendicular to AB and CD both.
Show that the straight lines whose direction cosines are given by 2l + 2m – n = 0 and mn + nl + lm = 0 are at right angles.
If l1, m1, n1; l2, m2, n2; l3, m3, n3 are the direction cosines of three mutually perpendicular lines, prove that the line whose direction cosines are proportional to l1 + l2 + l3, m1 + m2 + m3, n1 + n2 + n3 makes equal angles with them.
Distance of the point (α, β, γ) from y-axis is
If the directions cosines of a line are k,k,k, then
The distance of the plane from the origin is
The sine of the angle between the straight lineand the plane 2x – 2y + z = 5 is
The reflection of the point (α, β, γ) in the xy– plane is
The area of the quadrilateral ABCD, where A(0,4,1), B (2, 3, –1), C(4, 5, 0)and D (2, 6, 2), is equal to
The locus represented by xy + yz = 0 is
The plane 2x – 3y + 6z – 11 = 0 makes an angle sin–1(α) with x-axis. The valueof α is equal to
Fill in the blanks in each of the
A plane passes through the points (2,0,0) (0,3,0) and (0,0,4). The equation of plane is __________.
The direction cosines of the vectorare __________.
The vector equation of the lineis __________.
The vector equation of the line through the points (3,4,–7) and (1,–1,6) is__________.
The cartesian equation of the planeis __________.
State True or False for the statements
The unit vector normal to the plane x + 2 y +3z – 6 = 0 is.
The intercepts made by the plane 2x – 3y + 5z +4 = 0 on the co-ordinate axisare.
The angle between the lineand the plane is .
The angle between the planes and is .
The linelies in the plane
The vector equation of the line is
The equation of a line, which is parallel toand which passes through the point (5, –2,4), is.
If the foot of perpendicular drawn from the origin to a plane is (5, – 3, – 2),then the equation of plane is.
