DISTANCE
1) Find the distance from the origin to each of the points
A) (2,2,3). √17
B) (4,-1,2). √21
C) (0, 4, -4). 4√2
D) (-4, -3, -2). √29
2) Find the distance between each of the following pairs of points:
A) (2,5,3) and (-3,2,1). √38
B) (0,3,0) and (6,0,2). 7
C) (-4, -2,0) and (3,3,5). 3√11
3) Show that the triangle with vertices (6,10,10), (1,0,-1),(6,-10,0) is a right angled triangle, and find its area. 25√21sq,unit
4) Show that the triangle with vertices A(3,5-4), B(-1,1,2), C(-5,-5,-2) is isosceles.
5) Show that (4,2,4), (10,2-2) and (2,0,-4) are the vertices of an equilateral triangle.
6) Show that the points (1,-1,3), (2,-4,5) and (5,-13,11) are collinear.
7) Derive the equation of the locus of a point equidistant from the point (1,-2,3) and (-3,4,2). 8x-12y+2z+15= 0
8) Derive the equation of the locus of a point twice as far from (-2,3,4) as from (3,-1,-2). 3x²+3y²+3z²-28x+14y+24z+27=0
9) Find the equation of the locus of a point whose distance from the Y-axis is equal to its distance from (2,1,-1). y²-2y-4x+2z+6=0
10) Find the equation of the locus of a point whose distance from xy-plane is equals to its distance from the point (-1,2,-3). x²+y²+2x- 4y+ 6z+14= 0
11) A point moves so that the difference of the squares of its distances from the x-axis and the y-axis is constant. Find the equation of locus. y²- x² = a
12) Find the equation of the locus of a point whose distance from the z-axis is equal to its distance from the xy plane. x²+y²+z²= 0
DIVISION or SECTION Formula
1) Find the coordinates of the points which divide the join of the points (2,-1,3) and (4,3,1) in the ratio 3:4 internally. (20/7,5/7,15/7)
2) Find the Coordinates of the points which the line joining the points (2,-4,3), (-4,5,-6) in the ratio
A) 1:-4 (4,-7,6)
B) 2:1 (-2,2,-3)
3) Find the ratio in which the line joining the points (2,4,5),(3,5,-4) is divided by the yz plane. -2:3
4) The three points A(0,0,0), B(2,-3,3), C (-2,3,-3) are collinear. Find in which ratio each point divides the segment joining the other two. AB/BC= -1/2, BC/CA= -2/1, CA/AB=1/1.
5) Find the coordinates of the point which trisect AB given that (2,1,-3) and B(5,-8,3). (3,-2,1),(4,-5,1)
6) Find the coordinates of the point which is three-fifth of the way from (3,4,5) to (-2,-1,0). (0,1,2)
7) Show that the point (1,-1,2) is common to the lines which join (6,-7,0) to (16,-19 ,-4) and (0,3,-6) to (2,-5,10).
8) Find the lengths of the medians of the triangle whose vertices are A(2,-3,1), B(-6,5,3), C(8,7,-7). √91,√166,√217
9) Find the point of intersection of the medians of the triangle with vertices (-1,-3,-4), (4,-2,-7),(2,3,-8). (5/3,-2/3,-19/3)
10) Find the ratio in which the join of (2,1,5) and (3,4,3) is divided by the plane 2x+2y-2z=1, also, find the coordinates of the point of division. 5:7,(29/12,9/4,25/6)
11) The midpoints of the sides of a triangle (1,5,0),+0,4,-2) and (2,3,4). find its vertices. (1,2,3),(3,4,5)(-1,6,-7)
12) Three vertices of a parallelogram ABCD are A(3,-1,2), B(1,2,-4) and C(-1,1,2). Find the Coordinates of the fourth verex D. (1,-2,8)
13) What is the locus of a point for which
A) x= 0. yz- plane
B) y= 0. yz- plane
C) z= 0. xy- plane
D) x= a. The plane parallel to the yz- plane at a distance a unit from it.
E) y= b. The plane parallel to the xz plane at a distance b units from it.
F) z= c. The plane parallel to the xy plane at a distance c units from it.
14) What is the locus of a point for which.
A) x= 0, y= 0. The z-axis
B) y= 0, z=0. The x-axis
C) z= 0, x= 0. The y-axis
D) x= a, y= b. The line of intersection of the given planes x=a and y=b
E) y= b, z=c. The line of intersection of the given planes y=b and z=c
F) z= c, x= a. The line of intersection of the given planes z=c and x=a
15) Find the ratio in which the xy-plane divides the join of (-3,4,-8) and (5,-6,4). Also obtain the point of intersection of this line with the plane. 2:1, (7/3,-8/3,0)
16) Given that P(3,2-4), Q(5,4,-6), R(9,8,-10) are collinear, find the ratio in which Q divides PR. (38/16,57/16,17/16)
17) Using section formula, prove that the three points A(-2,3,5), B(1,2,3) and C(7,0,-1) are collinear.
18) Show that the point (1,2,3) is common to the lines which join A(4,8,12) to B(2,4,6) and (3,5,4) to D(5,8,5)
ANGLES, DIRECTION RATIOS, COSINES.....
1) The direction ratios of a line are 1, -2, -2. What are their direction cosines ? 1/3,-2/3,-2/3
2) If k, l, m are angles which a line makes with the axes, prove that sin²k+ sin²l+ sin²m= 2.
3) Can a line have direction and angles 45°,60°, 120° ?
4) Prove that 1, 1,1 cannot be direction cosines of a straight line.
5) Find the direction cosines and direction ratios of the line joining the points:
A) A(0,0 ,0), B(4,8,-8).
1/3,2/3,-2/3; 2, -2
B) A(1,3,5 ), B(-1,0-1).
-2/7, -3/7,-6/7; 2,3,6
C) A(5,6,-3), B(1,-6,3).
-2/7,-6/7,3/7; 2,6,-3
D) A(4,2,-6), B(-2,1,3).
6/√118, 1/√118, -9/√118; 6,1, -9
6) By using direction ratios method, show that following set of points are collinear.
A) A(1,2,3), B(4,0,4) and C(-2,4,2)
B) (-2,4,7),(3,-6,-8), (1,-2,-2)
7) A line makes an angle of π/4 with each of the x-axis and the Y-Axes. Find the angle made by it with the z-axis. π/2
8) If the line OP makes with x- axis an angle of measure 120° and with y-axis an angle of measure 60°. Find the angle made by the line with the Z-axis. 45° or 135°
9) Find the angle between the vectors whose direction cosines are proportional to 2,3,-6 and 3,-4,5.
cos⁻¹{(18√2)/35}
10) If k, l, m are the angles that line makes with the axes, then find cos m if
A) cos k= 14/15, cos l= -1/3. ±2/15
B) k= 60°, l,= 135° ±1/2
11) If the coordinates of A and B be (2,3,4 and (1,-2,1) respectively. prove that OA is perpendicular to OB, where O is the origin.
12) Show that the join of the points (1,2,3), (4,5,7) is parallel to the join the join the points (-4,3,-6) and (2,9,2).
13) Find the angles between the lines whose direction ratios are
A) 5,-12,13; -3,4,5 cos⁻¹(1/65)= 89°6'
B) 1,1,2; √3-1,-√3-1,4. π/3
14) If P,Q,R are respectively (2,3,5), (-1,3,2) and (3,5,-2). find the direction cosines of the sides of the of the sides of the of the triangle PQR. -2/3,-1/3,2/3; 1/3√6, 2/3√6, -7/3√6,; 1/√2, 0, 1/√2
15) Prove that three points P, Q ,R, whose coordinates are respectively (3,2,-4),(5,4,-6) and (9,8,-10) are collinear and find the ratio in which in which Q divides PR. 1:2
16) Find the angle not greater than 90° between the lines joining the following pairs of points:
A) (8,2,0),(4,6,-7), and (-3,1,2),
(-9,-2,4); 88°11'
B) (4,-2,3),(6,1,7) and (4,-2,3),(5,4,-2). 90°
C) (3,1,-2),(4,0,-4), and (4,-3,3),(6,-2,2). π/3
17) Find the direction cosines of a line which is perpendicular to the lines with directions cosines proportional to 1, -2, -2; 0, 2, 1.
2/3, -1/3, 2/3
18) Find the direction ratios of a perpendicular to the two lines determined by the pairs of points (2,3,-4),(-3,3,-2), and (-1,4,2),(3,5,1). -2,3,-5
19) For what value of x will the line through (4,1,2),(5,x,0) be parallel to the line through (2,1,1) and (3,3,-1).
x= 3
20) For what value of x the lines in the above(19) Problem be perpendicular? -3/2
21) show that the points (4,7,8), (2,3,4),(-1,-2,1) and (1,2,5) are the vertices of a Parallelogram.
22) Show that the points (5,-1,1),(7,-4,7),(1,-6,10) and (-1,-3,4) are the vertices of a rhombus.
23) Find the foot of the perpendicular drawn from the point A(1,0,3) to the join of the points B (4,7,1) and C(3,5,3) . 5/3,7/3,17/3
24) A(1,0,4) and B(0,-11,3), C(2,-3,1) are three points and D is the foot of the perpendicular from A on BC. Find the co-ordinate of D.
22/9,-11/9,5/9
25) Calculate the cosine of the angle A of the triangle with the vertices A(1,-1,2),B(6,11, 2), C(1, 2,6). 36/65
26) if A,B,C,D are the points (6,-6,0),(1,-7,6),(3,-4,4),(2,-9,2) respectively, prove that AB is perpendicular to CD.
27) Find the angle between any two diagonals of a cube. Cosk=1/3
MISCELLANEOUS QUESTIONS
1) If a line makes angles 90°,135°, 45° with the positive x,y and z Axis respectively. find its direction cosines. 0,-1/√2,1/√2
2) If a line has the direction ratios -18,12,-4, then what are the direction cosines? -9/11,6/11,-2/11
3) Find the distance between the points (7,4,-5) and (1,6,-2) show that these two points are collinear with the point (-5,8,1). 7 units
4) Find the ratio in which the join of (-3,5,6) and (4,6,-5) is divided by the yz-plane. 3:4
5) Show that the line joining the points A(7,4,2) and B(3,-2,5) is parallel to the line joining the points to C(2,-3,5) and D(-6,-15,11).
6) Find the angle between the lines whose direction ratios are 2,3,6 and 1,2,2. Cosk= 20/21
7) If A(6,-6,0), B(-1,-7,6), C(3,-4,4) and D(2,-9,2) be 4 points in space, show that AB perpendicular to CD.
8) Show that the point (0,7,10),(-1,6,6) and (4,9,6) form an isosceles right-angles triangle.
(Hints: show that angle between two sides = 90° by using the formula Cosk = l'.l"+ m'.m" + n'.n". Also, show that two angles are equal.)
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