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Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, −1), (4, 3, −1).
Let OA be the line joining the origin, O (0, 0, 0), and the point, A (2, 1, 1).
Also, let BC be the line joining the points, B (3, 5, −1) and C (4, 3, −1).
The direction ratios of OA are 2, 1, and 1 and of BC are (4 − 3) = 1, (3 − 5) = −2, and (−1 + 1) = 0
OA is perpendicular to BC, if a1a2 + b1b2 + c1c2 = 0
∴ a1a2 + b1b2 + c1c2 = 2 × 1 + 1 (−2) + 1 ×0 = 2 − 2 = 0
Thus, OA is perpendicular to BC.
If l1, m1, n1 and l2, m2, n2 are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m1n2 − m2n1, n1l2 − n2l1, l1m2 − l2m1.
It is given that l1, m1, n1 and l2, m2, n2 are the direction cosines of two mutually perpendicular lines. Therefore,
Let l, m, n be the direction cosines of the line which is perpendicular to the line with direction cosines l1, m1, n1 and l2, m2, n2.
l, m, n are the direction cosines of the line.
∴l2 + m2 + n2 = 1 … (5)
It is known that,
∴ 
Substituting the values from equations (5) and (6) in equation (4), we obtain
Thus, the direction cosines of the required line are 
Find the angle between the lines whose direction ratios are a, b, c and b − c,
c − a, a − b.
The angle Q between the lines with direction cosines, a, b, c and b − c, c − a,
a − b, is given by,
Thus, the angle between the lines is 90°.
Find the equation of a line parallel to x-axis and passing through the origin.
The line parallel to x-axis and passing through the origin is x-axis itself.
Let A be a point on x-axis. Therefore, the coordinates of A are given by (a, 0, 0), where a ∈ R.
Direction ratios of OA are (a − 0) = a, 0, 0
The equation of OA is given by,
Thus, the equation of line parallel to x-axis and passing through origin is
If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (−4, 3, −6) and (2, 9, 2) respectively, then find the angle between the lines AB and CD.
The coordinates of A, B, C, and D are (1, 2, 3), (4, 5, 7), (−4, 3, −6), and
(2, 9, 2) respectively.
The direction ratios of AB are (4 − 1) = 3, (5 − 2) = 3, and (7 − 3) = 4
The direction ratios of CD are (2 −(− 4)) = 6, (9 − 3) = 6, and (2 −(−6)) = 8
It can be seen that, 
Therefore, AB is parallel to CD.
Thus, the angle between AB and CD is either 0° or 180°.
If the lines 
and 
are perpendicular, find the value of k.
The direction of ratios of the lines, and , are −3, 2k, 2 and 3k, 1, −5 respectively.
It is known that two lines with direction ratios, a1, b1, c1 and a2, b2, c2, are perpendicular, if a1a2 + b1b2 + c1c2 = 0
Therefore, for
, the given lines are perpendicular to each other.
Find the vector equation of the line passing through (1, 2, 3) and perpendicular to the plane
The position vector of the point (1, 2, 3) is 
The direction ratios of the normal to the plane, 
, are 1, 2, and −5 and the normal vector is 
The equation of a line passing through a point and perpendicular to the given plane is given by, 
Find the equation of the plane passing through (a, b, c) and parallel to the plane 
Any plane parallel to the plane, 
, is of the form 
The plane passes through the point (a, b, c). Therefore, the position vector 
of this point is 
Therefore, equation (1) becomes
Substituting 
in equation (1), we obtain
This is the vector equation of the required plane.
Substituting 
in equation (2), we obtain
Find the shortest distance between lines 
and
.
The given lines are
It is known that the shortest distance between two lines, 
and 
, is given by
Comparing 
to equations (1) and (2), we obtain
Substituting all the values in equation (1), we obtain
Therefore, the shortest distance between the two given lines is 9 units.
Find the coordinates of the point where the line through (5, 1, 6) and
(3, 4, 1) crosses the YZ-plane
It is known that the equation of the line passing through the points, (x1, y1, z1) and (x2, y2, z2), is 
The line passing through the points, (5, 1, 6) and (3, 4, 1), is given by,
Any point on the line is of the form (5 − 2k, 3k + 1, 6 −5k).
The equation of YZ-plane is x = 0
Since the line passes through YZ-plane,
5 − 2k = 0
Therefore, the required point is 
.
Find the coordinates of the point where the line through (5, 1, 6) and
(3, 4, 1) crosses the ZX − plane.
It is known that the equation of the line passing through the points, (x1, y1, z1) and (x2, y2, z2), is 
The line passing through the points, (5, 1, 6) and (3, 4, 1), is given by,
Any point on the line is of the form (5 − 2k, 3k + 1, 6 −5k).
Since the line passes through ZX-plane,
Therefore, the required point is
.
Find the coordinates of the point where the line through (3, −4, −5) and (2, − 3, 1) crosses the plane 2x + y + z = 7).
It is known that the equation of the line through the points, (x1, y1, z1) and (x2, y2, z2), is 
Since the line passes through the points, (3, −4, −5) and (2, −3, 1), its equation is given by,
Therefore, any point on the line is of the form (3 − k, k − 4, 6k − 5).
This point lies on the plane, 2x + y + z = 7
∴ 2 (3 − k) + (k − 4) + (6k − 5) = 7
Hence, the coordinates of the required point are (3 − 2, 2 − 4, 6 × 2 − 5) i.e.,
(1, −2, 7).
Find the equation of the plane passing through the point (−1, 3, 2) and perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0.
The equation of the plane passing through the point (−1, 3, 2) is
a (x + 1) + b (y − 3) + c (z − 2) = 0 … (1)
where, a, b, c are the direction ratios of normal to the plane.
It is known that two planes, 
and 
, are perpendicular, if 
Plane (1) is perpendicular to the plane, x + 2y + 3z = 5
Also, plane (1) is perpendicular to the plane, 3x + 3y + z = 0
From equations (2) and (3), we obtain
Substituting the values of a, b, and c in equation (1), we obtain
This is the required equation of the plane.
If the points (1, 1, p) and (−3, 0, 1) be equidistant from the plane 
, then find the value of p.
The position vector through the point (1, 1, p) is 
Similarly, the position vector through the point (−3, 0, 1) is
The equation of the given plane is 
It is known that the perpendicular distance between a point whose position vector is 
and the plane, 
is given by, 
Here,
and d
Therefore, the distance between the point (1, 1, p) and the given plane is
Similarly, the distance between the point (−3, 0, 1) and the given plane is
It is given that the distance between the required plane and the points, (1, 1, p) and (−3, 0, 1), is equal.
∴ D1 = D2
Find the equation of the plane passing through the line of intersection of the planes 
and 
and parallel to x-axis.
The given planes are
The equation of any plane passing through the line of intersection of these planes is
[r→.(i?+j?+k?)−1]+λ[r→.(2i?+3j?−k?)+4]=0r→.i?+j?+k?-1+λr→.2i?+3j?-k?+4=0
r→.[(2λ+1)i?+(3λ+1)j?+(1−λ)k?]+(4λ−1)=0 ...(1)r→.2λ+1i?+3λ+1j?+1-λk?+4λ-1=0 ...(1)
Its direction ratios are (2λ + 1), (3λ + 1), and (1 − λ).
The required plane is parallel to x-axis. Therefore, its normal is perpendicular to x-axis.
The direction ratios of x-axis are 1, 0, and 0.
Substituting 
in equation (1), we obtain
Therefore, its Cartesian equation is y − 3z + 6 = 0
This is the equation of the required plane.
If O be the origin and the coordinates of P be (1, 2, −3), then find the equation of the plane passing through P and perpendicular to OP.
The coordinates of the points, O and P, are (0, 0, 0) and (1, 2, −3) respectively.
Therefore, the direction ratios of OP are (1 − 0) = 1, (2 − 0) = 2, and (−3 − 0) = −3
It is known that the equation of the plane passing through the point (x1, y1z1) is
where, a, b, and c are the direction ratios of normal.
Here, the direction ratios of normal are 1, 2, and −3 and the point P is (1, 2, −3).
Thus, the equation of the required plane is
Find the equation of the plane which contains the line of intersection of the planes 
, 
and which is perpendicular to the plane
.
The equations of the given planes are
The equation of the plane passing through the line intersection of the plane given in equation (1) and equation (2) is
The plane in equation (3) is perpendicular to the plane, 
Substituting 
in equation (3), we obtain
This is the vector equation of the required plane.
The Cartesian equation of this plane can be obtained by substituting 
in equation (3).
Find the distance of the point (−1, −5, −10) from the point of intersection of the line 
and the plane
.
The equation of the given line is
The equation of the given plane is
Substituting the value of 
from equation (1) in equation (2), we obtain
Substituting this value in equation (1), we obtain the equation of the line as
This means that the position vector of the point of intersection of the line and the plane is
This shows that the point of intersection of the given line and plane is given by the coordinates, (2, −1, 2). The point is (−1, −5, −10).
The distance d between the points, (2, −1, 2) and (−1, −5, −10), is
Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes 
and 
.
Let the required line be parallel to vector 
given by,
The position vector of the point (1, 2, 3) is 
The equation of line passing through (1, 2, 3) and parallel to is given by,
The equations of the given planes are
The line in equation (1) and plane in equation (2) are parallel. Therefore, the normal to the plane of equation (2) and the given line are perpendicular.
From equations (4) and (5), we obtain
Therefore, the direction ratios of are −3, 5, and 4.
Substituting the value of in equation (1), we obtain
This is the equation of the required line.
Find the vector equation of the line passing through the point (1, 2, − 4) and perpendicular to the two lines: 
Let the required line be parallel to the vector 
given by, 
The position vector of the point (1, 2, − 4) is 
The equation of the line passing through (1, 2, −4) and parallel to vector is
The equations of the lines are
Line (1) and line (2) are perpendicular to each other.
Also, line (1) and line (3) are perpendicular to each other.
From equations (4) and (5), we obtain
∴Direction ratios of are 2, 3, and 6.
Substituting 
in equation (1), we obtain
This is the equation of the required line.
Prove that if a plane has the intercepts a, b, c and is at a distance of P units from the origin, then 
The equation of a plane having intercepts a, b, c with x, y, and z axes respectively is given by,
The distance (p) of the plane from the origin is given by,
Distance between the two planes: 
and 
is
(A)2 units (B)4 units (C)8 units
(D)
The equations of the planes are
It can be seen that the given planes are parallel.
It is known that the distance between two parallel planes, ax + by + cz = d1 and ax + by + cz = d2, is given by,
Thus, the distance between the lines is 
units.
Hence, the correct answer is D.
The planes: 2x − y + 4z = 5 and 5x − 2.5y + 10z = 6 are
(A) Perpendicular (B) Parallel (C) intersect y-axis
(C) passes through 
The equations of the planes are
2x − y + 4z = 5 … (1)
5x − 2.5y + 10z = 6 … (2)
It can be seen that,
∴ 
Therefore, the given planes are parallel.
Hence, the correct answer is B.
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