EXERCISE-1
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2) Prove by using distance formula that the points P(1,2,3), Q(-1,-1,-1) and R(3,5,7) are collinear.
3) Determine the point on XY- plane which is equidistant from three points A(2,0,3), B(0,3,2) and (0,0,1). (3,2,0)
4) Show that the point A(0,1,2), B(2,-1,3) and C(1,-3,1) are vertices of an isosceles right angle triangle.
5) Find the locus of the point of point which is equidistant from the points A(0,2,3) and B(2,-2,1). x - 2y - z+ 1= 0
6) Find the coordinates of a point equidistant from the four points O(0, 0,0), A(a,0,0), B(0,b,0) and C(0, 0, c). (a/2,b/2,c/2)
7) find the coordinates of the point which divides the join of P(2,1,4) and Q(4,3,2) in the ratio 2:3
A) internally. (14/5,3/5,16/5)
B) externally. (-2,-9,8)
8) Find the ratio in which the line joining the point (1,2,3) and (-3,4,-5) is divided by the xy plane. also, find the coordinates of the point of division. 3:5, (-1/2,11/4,0)
9) find the ratio in which join the A(2,1,5) and B(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)
10) Using Section formula, prove that three points A(-2,3,5), B(1,2,3) and C(7,0, -1) are collinear.
11) The midpoints of the sides of a triangle are (1,5,-1), (0,4,-2) and (2,3,4). Find its vertices. (1,2,3),(3,4,5),(-1,6,-7)
12) Given that P(3,2,-4), Q(5,4,-6) and R(9,8,-10) are collinear. Find the ratio in which Q divides PR. 1:2
13) Find the coordinates of the points which trisect the line segment AB, given that A(2,1,-3) and B(5,-8,3). (4,-5,1)
14) Find the coordinates of the foot of the perpendicular drawn from the point A(1,2,1) to the line joining B(1,4,6) and (5,4,4). (3,4,5)
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** NOTE:
1) DIRECTION COSINES:
The direction Cosines of a line are defined as the direction Cosines of any vector whose support is the given line.
Direction Cosines are either cos a, cos b, cos c OR -cos a, -cos b, -cos c.
Therefore, if l, m, n are Direction Cosines of a line, then - l, - m, - n are also its Direction Cosines and always have l²+ m² + n²= 1.
If A(x₁, y₁, z₁) and B (x₂, y₂, z₂) are two points on a line L, then its Direction Cosines are:
(x₂ - x₁)/AB , (y₂ - y₁)/AB, (z₂ - z₁)/AB OR
(x₁- x₂)/AB, (y₁-y₂)/AB, (z₁ - z₂)/AB
2) DIRECTION RATIOS:
The direction Ratios of a line are proportional to the direction Ratios of any vector whose support is the given line.
If A(x₁, y₁, z₁) and B (x₂, y₂, z₂) are two points on a line, then its Direction Ratios are proportional to:
x₂ - x₁, y₂ - y₁ , z₂ - z₁.
3) ANGLE BETWEEN TWO VECTORS:
It is defined as the angle between two vectors parallel to them. So, the results derived for vectors will also be applicable to lines. ₁₂ ₁ₓ ₁₂
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EXERCISE-2
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1) Using vector method: prove that the points A(3,-2,4), B(1,1,1) and C(-1,4,2) are collinear.
2) Find the distance between the points A and B with position vectors i - j and 2i+ j + 2k. 3
3) Find the angle between the vectors with direction ratios proportional to 4, -3, 5 & 3,4,5. π/3
4) find the angle between the lines whose direction ratios are proportional to 4, -3, 5 & 3, 4, 5. π/3
5) P(6,3,2), Q(5, 1,4) and R(3, 3,5) are the vertices of a triangle PQR. Find ang.PQR. π/2
6) Find the coordinates of the foot of the perpendicular drawn from the point A(1,2,1) to the line joining B(1,4,6) and C(5,4,4). (3,4,5)
7) find the direction cosines of the line which is perpendicular to the lines with direction cosines proportional to 1, -2 2 and 0,2,1. 2/3,-1/3,2/3
8) find the direction cosines of the sides of the triangle whose vertices are (3,5,-4), (-1,1,2) and (-5, -5,-2) and also find the angles of the triangle, what types of triangle it is ? Isosceles obtuse angled triangle
9) find the angle between the lines whose direction cosines are given by the equations 3l+ m + 5n= 0, 6mn- 2nl+ 5lm = 0. Cos⁻¹ (-1/6)
10) Find the direction cosines of the two lines which are connected by the relations. l -5m + 3n= 0 and 7l²- 3n²= 0. ±-1/6, ±1/6, ± 2/√6
EXERCISE -3
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1) If a line makes angles of 90°, 60°and 30°with the positive direction of x, y and z-Axis. respectively, find the direction cosines. 0, 1/2, √3/2
2) If a line and direction ratios 2, -1, 2, determine its direction cosines. 2/3,-1/3,-2/3
3) find the direction cosines of the line passing through two points (-2, 4, -5) and (1,2, 3). 3/√77, -2/√77, 8√77
4) Using direction ratios shows that the points A(2, 3, -4) and B(1, -2, 3) and C(3,8, -11) are collinear.
5) Find the direction cosines of the sides of the triangle whose vertices are (3,5,-4), (-1,1,2) and (-5,-5,-2). 2/√17, 2√17, -3/√17, ; 2/√17, 3/√17, 2/√17; 4/√42, 5/√42, -1/√42
6) Find the angle between the vectors with direction ratios proportional to 1,-2,1 and 4,3,2. π/2
7) find the angle between the vectors whose direction cosines are proportional to 2, 3, -6 and 3,-4,5. Cos⁻¹{-(18√2)/35}
8) Find the acute angle between the lines whose direction ratios are proportional to 2:3:6 and 1:2:2. Cos⁻¹(20/21)
9) Show that the point (2,3,4), (-1,-2,1), (5,8,7) are collinear.
10) Show that the line through the points (4,7,8) and (2,3,4) is parallel to the line through the points (-1,-2,1) and (1,2,5).
11) show that the line through the points (1,-1,2) and (3,4,-2) is perpendicular to the line through the points (0,3,2) and (3,5,6).
12) 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) and (4,3,-1).
13) Find the angle between the lines whose direction ratios are proportional to a, b, c and b- c, c - a, a - b. π/2
14) If the coordinates of the points A, B, C , D are (1,2,3), (4,5,7), (-4,3,-6) and (2,9,2), then find the angle between AB and CD. 0
15) Find the direction cosines of the lines, connected by the relations; l+ m+ n= 0 and 2lm + 2ln - mn= 0. ±1/√6, ±1/√6, ± -2/56; ±-1/√6, ±2/√6, ±1/√6.
16) find the angle between the lines whose direction cosines are given by equations:
A) l+ m+ n= 0 and l² + m² - n²= 0
B) 2l- m+ 2n= 0 and mn + ln + ml= 0.
C) l+ 2m+ 3n= 0 and 3lm - 4ln + mn= 0.
D) 2l+ 2m- n= 0 and lm +ln + mn= 0. (π/3, π/2, π/2, π/2)
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