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Show that the three lines with direction cosines
are mutually perpendicular.
Two lines with direction cosines, l1, m1, n1 and l2, m2, n2, are perpendicular to each other, if l1l2 + m1m2 + n1n2 = 0
(i) For the lines with direction cosines, 
and 
, we obtain
Therefore, the lines are perpendicular.
(ii) For the lines with direction cosines, and 
, we obtain
Therefore, the lines are perpendicular.
(iii) For the lines with direction cosines, and , we obtain
Therefore, the lines are perpendicular.
Thus, all the lines are mutually perpendicular.
Show that the line through the points (1, −1, 2) (3, 4, −2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).
Let AB be the line joining the points, (1, −1, 2) and (3, 4, − 2), and CD be the line joining the points, (0, 3, 2) and (3, 5, 6).
The direction ratios, a1, b1, c1, of AB are (3 − 1), (4 − (−1)), and (−2 − 2) i.e., 2, 5, and −4.
The direction ratios, a2, b2, c2, of CD are (3 − 0), (5 − 3), and (6 −2) i.e., 3, 2, and 4.
AB and CD will be perpendicular to each other, if a1a2 + b1b2+ c1c2 = 0
a1a2 + b1b2+ c1c2 = 2 × 3 + 5 × 2 + (− 4) × 4
= 6 + 10 − 16
= 0
Therefore, AB and CD are perpendicular to each other.
Show that the line through the points (4, 7, 8) (2, 3, 4) is parallel to the line through the points (−1, −2, 1), (1, 2, 5).
Let AB be the line through the points, (4, 7, 8) and (2, 3, 4), and CD be the line through the points, (−1, −2, 1) and (1, 2, 5).
The directions ratios, a1, b1, c1, of AB are (2 − 4), (3 − 7), and (4 − 8) i.e., −2, −4, and −4.
The direction ratios, a2, b2, c2, of CD are (1 − (−1)), (2 − (−2)), and (5 − 1) i.e., 2, 4, and 4.
AB will be parallel to CD, if 
Thus, AB is parallel to CD.
Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector
.
It is given that the line passes through the point A (1, 2, 3). Therefore, the position vector through A is 
It is known that the line which passes through point A and parallel to 
is given by 
is a constant.
This is the required equation of the line.
Find the equation of the line in vector and in Cartesian form that passes through the point with position vector 
and is in the direction 
.
It is given that the line passes through the point with position vector
It is known that a line through a point with position vector 
and parallel to 
is given by the equation, 
This is the required equation of the line in vector form.
Eliminating λ, we obtain the Cartesian form equation as
This is the required equation of the given line in Cartesian form.
Find the Cartesian equation of the line which passes through the point
(−2, 4, −5) and parallel to the line given by
It is given that the line passes through the point (−2, 4, −5) and is parallel to
The direction ratios of the line, , are 3, 5, and 6.
The required line is parallel to
Therefore, its direction ratios are 3k, 5k, and 6k, where k ≠ 0
It is known that the equation of the line through the point (x1, y1, z1) and with direction ratios, a, b, c, is given by 
Therefore the equation of the required line is
The Cartesian equation of a line is 
. Write its vector form.
The Cartesian equation of the line is
The given line passes through the point (5, −4, 6). The position vector of this point is 
Also, the direction ratios of the given line are 3, 7, and 2.
This means that the line is in the direction of vector, 
It is known that the line through position vector 
and in the direction of the vector 
is given by the equation, 
This is the required equation of the given line in vector form.
Find the vector and the Cartesian equations of the lines that pass through the origin and (5, −2, 3).
The required line passes through the origin. Therefore, its position vector is given by,
The direction ratios of the line through origin and (5, −2, 3) are
(5 − 0) = 5, (−2 − 0) = −2, (3 − 0) = 3
The line is parallel to the vector given by the equation, 
The equation of the line in vector form through a point with position vector 
and parallel to 
is, 
The equation of the line through the point (x1, y1, z1) and direction ratios a, b, c is given by, 
Therefore, the equation of the required line in the Cartesian form is
Find the vector and the Cartesian equations of the line that passes through the points (3, −2, −5), (3, −2, 6).
Let the line passing through the points, P (3, −2, −5) and Q (3, −2, 6), be PQ.
Since PQ passes through P (3, −2, −5), its position vector is given by,
The direction ratios of PQ are given by,
(3 − 3) = 0, (−2 + 2) = 0, (6 + 5) = 11
The equation of the vector in the direction of PQ is
The equation of PQ in vector form is given by, 
The equation of PQ in Cartesian form is
i.e.,
Find the angle between the following pairs of lines:
(i) 
(ii) 
and
(i) Let Q be the angle between the given lines.
The angle between the given pairs of lines is given by, 
The given lines are parallel to the vectors, 
and 
, respectively.
(ii) The given lines are parallel to the vectors, 
and 
, respectively.
Find the angle between the following pairs of lines:
(i) 
(ii) 
Let 
and 
be the vectors parallel to the pair of lines, , respectively.
and 
The angle, Q, between the given pair of lines is given by the relation,
2. Let 
be the vectors parallel to the given pair of lines, 
and 
, respectively.
If Q is the angle between the given pair of lines, then 
Find the values of p so the line 
and
are at right angles.
The given equations can be written in the standard form as
and 
The direction ratios of the lines are −3,
, 2 and 
respectively.
Two lines with direction ratios, a1, b1, c1 and a2, b2, c2, are perpendicular to each other, if a1a2 + b1 b2 + c1c2 = 0
Thus, the value of p is 
.
Show that the lines 
and 
are perpendicular to each other.
The equations of the given lines areand
The direction ratios of the given lines are 7, −5, 1 and 1, 2, 3 respectively.
Two lines with direction ratios, a1, b1, c1 and a2, b2, c2, are perpendicular to each other, if a1a2 + b1 b2 + c1c2 = 0
∴ 7 × 1 + (−5) × 2 + 1 × 3
= 7 − 10 + 3
= 0
Therefore, the given lines are perpendicular to each other.
Find the shortest distance between the lines
The equations of the given lines are
It is known that the shortest distance between the lines, 
and 
, is given by,
d = ????(b1→×b2→).(a2→−a1→)???b1→×b2→???????d = b1→×b2→.a2→-a1→b1→×b2→
Comparing the given equations, we obtain
Substituting all the values in equation (1), we obtain
Therefore, the shortest distance between the two lines is 
units.
Find the shortest distance between the lines 
and 
The given lines are and
It is known that the shortest distance between the two lines, 
, is given by,
Comparing the given equations, we obtain
Substituting all the values in equation (1), we obtain
Since distance is always non-negative, the distance between the given lines is 
units.
Find the shortest distance between the lines whose vector equations are
The given lines are 
and 
It is known that the shortest distance between the lines, 
and 
, is given by,
Comparing the given equations with and 
, we obtain 
Substituting all the values in equation (1), we obtain
Therefore, the shortest distance between the two given lines is 
units.
Find the shortest distance between the lines whose vector equations are
The given lines are and
It is known that the shortest distance between the lines, and , is given by,
Comparing the given equations with and , we obtain
Substituting all the values in equation (1), we obtain
Therefore, the shortest distance between the two given lines is units.
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