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If a line makes angle 90°, 135°, 45° with the positive x, y and z-axis respectively, find its direction cosines.
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Find the direction cosines of a line which makes equal angles with the coordinate axes.
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If a line has
direction cosines
then
what are its ratios ?
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If a line has the direction ratios –18, 12, –4, then what are its direction cosines ?
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Show that the points (2, 3, 4), (–1, –2, 1), (5, 8, 7) are collinear.
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Find the equation of the line which passes through the point
(1, 2, 3) and is parallel to the vector
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Find
the equation of the lien is vector and in Cartesian form that passes through
the point with positive vector
and
is in the direction
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Find the Cartesian equation of the line which passes through
the point (?2, 4, ? 5) and parallel to the line graph by
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The
cartesian equation of a line is
write
its vector form.
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Find the vector and the Cartesian equations of the lines that passes through the origin and (5, –2, 3).
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Find
the angle between the following pair of lines :
(i)
and
(ii)
and
(iii)
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Find
the values of p so that the lines
are at right
angles.
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Show
that the lines
are
perpendicular to each other.
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Find
the shortest distance between the lines
and
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Find
the shortest distance between the lines
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Find
the shortest distance between the lines whose vector equations are
and
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Find
the shortest distance between the lines whose vector equations are
and
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Find the equations of the lines joining the following pair of vertices and then find the shortest distance between the lines:
(i) (0, 0,0), (1, 0, 2), (ii) (1, 3, 0), (0, 3, 0)
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In each of the following, determine the direction cosines of the normal to the plane and the distance from the origin:
(a) z = 2 (b) x + y + z = 1 (c) 2x + 3y – z = 5
(d) 5y + 8 = 0
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Find the vector equation of a plane which is a at a
distance of 7 units from the origin and which is normal to the vector
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Find
the Cartesian equation of the following planes
(a)
(b)
(c)
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In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
(a) 2x + 3y + 4z – 12 = 0
(b) 3y + 4z – 6 = 0
(c) z + y + z = 1
(d) 5y + 8 = 0
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Find
the vector and Cartesian equation of the planes.
(a)
that passes through
the point (1, 0, ?2) and the normal to the plane is
(b)
that passes through
the point (1, 4, 6) and the normal vector to the plane is
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Find the equation of the planes that pass through the sets of three points
(a) (1, 1, –1), (6, 4, –5), (–4, –2, 3)
(b) (1, 1, 0), (1, 2, 1), (–2, 2, –1)
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Find the intercepts cut off by the plane 2x + y – z = 5.
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Find the
equation of the plane with intercept 3 on the y-axis and parallel to ZOX plane.
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Find the equation of the plane through the intersection of the planes 3x –y + 2z – 4 = 0 and x + y + z – 2 = 0 is and the point (2,2,1).
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Find the vector equation of the plane passing through the
intersection of the planes
and
the point (2, 1, 3)
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Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0.
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Find the angle between the planes whose vector equation are
and
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In the following exercise determine whether the given planes are parallel to perpendicular and in case they are neither, find the angle between them.
(a) 7x + 5y + 6z + 30 = 0
and 3x – y – 10z + 4 = 0
(b) 2x + y + 3z – 2 = 0 and x – 2y + 5 = 0
(c) 2x – 2y + 4z + 5 = 0 and 3x – 3y + 6z – 1 = 0
(d) 2x – y + 3z – 1 = 0 and 2x – y + 3z + 3 = 0
(e) 4x + 8y + z – 8 = 0 and y + z – 4 = 0
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In the following exercise, find the distance of each of the given points from the corresponding given plane.
Points Planes
(i) (0, 0, 0) 3x – 4y + 12z = 3
(ii) (3, –2, 1) 2x – y + 2z + 3 = 0
(iii) (2, 3, –5) x + 2y – 2z = 9
(iv) (–6, 0, 0) 2x – 3y + 6z – 2 = 0
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Show that the line joining the origin to the plane (2, 1, 1) is perpendicular to the line determined by the points (3, 5, –1), (4, 3, –1).
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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 – n1l1 ; l1 m1 – l2m1.
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Find the angle between the lines whose direction ratios are a, b, c and b –c, c–a, a–b.
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Find the angle between the lines whose direction ratios are a, b, c and b –c, c–a, a–b.
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Find the equation of a line parallel to X-axis and passing through the origin.
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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.
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If the lines
perpendicular, then find the value of K.
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Find
the vector equation of the straight line passing through (1, 2, 3) and
perpendicular to the plane
.
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Find the equation of the plane passing through (a, b, c) and
parallel to the plane
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Find
the shortest distance between lines
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Find the coordinates of the points where the line through (5, 1, 6) and (3, 4, 1) crosses the yz-plane.
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Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the zx-plane.
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Find the coordinates of the point where the line through (3, –4, –5) and (2, –3, 1) crosses the plane 2x + y + z = 7
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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.
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If
the points (1, 1, p) and (?3, 0, 1) be equidistant rom the plane
then find
the values of p.
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Find the equation of
the plane passing through the line of intersection of the planes
and parallel to
X-axis.
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If O be the origin and the coordinates of P and (1, 2, –3) the find the equation of the plane passing through P and perpendicular to OP.
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Find
the equation of the plane which contains the line of intersection of the planes
and
which is perpendicular to the plane
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Find
the distance of the point (?1, ?5, ?10) from the point of intersection of the
line
and the plane
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Find
the vector equation of the line passing through
(1,
2, 3) and parallel to the planes
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Find
the vector equation of the line passing through the point (1, 2, ?4) and
perpendicular to the two lines
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Show that the lines x = ay + b, z = cy + d and x = a’y + b’, z = c’y + d’ are perpendicular to each other, if aa’ + cc’ + 1 + 0.
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Prove
that if a plane has the intercepts a, b, c and is at a distance of p units from
the origin, then
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Equation of xz-plane is
(a) x = 0 (b) y = 0 (c) z = 0 (d) x = 0, z = 0
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Equation of the y-axis is
(a) x = 0 (b) y = 0; z = 0
(c) z = 0 (d) x = 0; z = 0
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Distance
between the two planes 2x = 3y + 4z = 4 and 4x + 45y + 8z = 12 is
(a)
2 units (b) 4 untis (c) 8 units
units.
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The
planes 2x ? y + 4z = 5 and 5x ? 2.5y + 10z = 6 are
(a)
perpendicular (b)
parallel (c) intersects along y-axis (d) passes through
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Vector
equation of a line that is parallel to X-axis is
(a)
(b)
(c)
(d)
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