Page No 27.23:
Question 1:
If a line makes angles of 90°, 60° and 30° with the positive direction of x, y, and z-axis respectively, find its direction cosines.
Answer:
Let the direction cosines of the line be l, m, n.
Now,
Page No 27.23:
Question 2:
If a line has direction ratios 2, −1, −2, determine its direction cosines.
Answer:
Page No 27.23:
Question 3:
Find the direction cosines of the line passing through two points (−2, 4, −5) and (1, 2, 3).
Answer:
Page No 27.23:
Question 4:
Using direction ratios show that the points A (2, 3, −4), B (1, −2, 3) and C (3, 8, −11) are collinear.
Answer:
Page No 27.23:
Question 5:
Find the direction cosines of the sides of the triangle whose vertices are (3, 5, −4), (−1, 1, 2) and (−5, −5, −2).
Answer:
Page No 27.23:
Question 6:
Find the angle between the vectors with direction ratios proportional to 1, −2, 1 and 4, 3, 2.
Answer:
Page No 27.23:
Question 7:
Find the angle between the vectors whose direction cosines are proportional to 2, 3, −6 and 3, −4, 5.
Answer:
Page No 27.23:
Question 8:
Find the acute angle between the lines whose direction ratios are proportional to 2 : 3 : 6 and 1 : 2 : 2.
Answer:
Page No 27.23:
Question 9:
Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
Answer:
Page No 27.23:
Question 10:
Show that the line through points (4, 7, 8) and (2, 3, 4) is parallel to the line through the points (−1, −2, 1) and (1, 2, 5).
Answer:
Page No 27.23:
Question 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).
Answer:
Thus, 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).
Page No 27.23:
Question 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).
Answer:
Therefore, 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).
Page No 27.23:
Question 13:
Find the angle between the lines whose direction ratios are proportional to a, b, c and b − c, c − a, a − b.
Answer:
Page No 27.23:
Question 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.
Answer:
Page No 27.23:
Question 15:
Find the direction cosines of the lines, connected by the relations: l + m +n = 0 and 2lm + 2ln − mn = 0.
Answer:
Page No 27.23:
Question 16:
Find the angle between the lines whose direction cosines are given by the equations
(i) l + m + n = 0 and l2 + m2 − n2 = 0
(ii) 2l − m + 2n = 0 and mn + nl + lm = 0
(iii) l + 2m + 3n = 0 and 3lm − 4ln + mn = 0
(iv) 2l + 2m − n = 0, mn + ln + lm = 0
Answer:
(iv) The given relations are
2l + 2m − n = 0 .....(1)
mn + ln + lm = 0 .....(2)
From (1), we have
n = 2l + 2m
Putting this value of n in (2), we get
When , we have
When , we have
Thus, the direction ratios of two lines are proportional to
and
Or and
So, vectors parallel to these lines are and .
Let be the angle between these lines, then is also the angle between and .
Thus, the angle between the two lines whose direction cosines are given by the given relations is .
Page No 27.24:
Question 1:
Define direction cosines of a directed line.
Answer:
Page No 27.24:
Question 2:
What are the direction cosines of X-axis?
Answer:
Page No 27.24:
Question 3:
What are the direction cosines of Y-axis?
Answer:
Page No 27.24:
Question 4:
What are the direction cosines of Z-axis?
Answer:
Page No 27.24:
Question 5:
Write the distances of the point (7, −2, 3) from XY, YZ and XZ-planes.
Answer:
Page No 27.24:
Question 6:
Write the distance of the point (3, −5, 12) from X-axis?
Answer:
Page No 27.24:
Question 7:
Write the ratio in which YZ-plane divides the segment joining P (−2, 5, 9) and Q (3, −2, 4).
Answer:
Page No 27.24:
Question 8:
A line makes an angle of 60° with each of X-axis and Y-axis. Find the acute angle made by the line with Z-axis.
Answer:
Page No 27.25:
Question 9:
If a line makes angles α, β and γ with the coordinate axes, find the value of cos 2α + cos 2β + cos 2γ.
Answer:
Page No 27.25:
Question 10:
Write the ratio in which the line segment joining (a, b, c) and (−a, −c, −b) is divided by the xy-plane.
Answer:
Page No 27.25:
Question 11:
Write the inclination of a line with Z-axis, if its direction ratios are proportional to 0, 1, −1.
Answer:
Page No 27.25:
Question 12:
Write the angle between the lines whose direction ratios are proportional to 1, −2, 1 and 4, 3, 2.
Answer:
Page No 27.25:
Question 13:
Write the distance of the point P (x, y, z) from XOY plane.
Answer:
Page No 27.25:
Question 14:
Write the coordinates of the projection of point P (x, y, z) on XOZ-plane.
Answer:
The projection of the point P (x, y, z) on XOZ-plane is (x, 0, z) as Y-coordinates of any point on XOZ-plane are equal to zero.
Page No 27.25:
Question 15:
Write the coordinates of the projection of the point P (2, −3, 5) on Y-axis.
Answer:
The coordinates of the projection of the point P ( 2, -3, 5) on the y-axis are ( 0, 3, 0) as both X and Z coordinates of each point on the y-axis are equal to zero.
Page No 27.25:
Question 16:
Find the distance of the point (2, 3, 4) from the x-axis.
Answer:
Page No 27.25:
Question 17:
If a line has direction ratios proportional to 2, −1, −2, then what are its direction consines?
Answer:
Page No 27.25:
Question 18:
Write direction cosines of a line parallel to z-axis.
Answer:
Page No 27.25:
Question 19:
If a unit vector makes an angle and an acute angle θ with , then find the value of θ.
Answer:
Page No 27.25:
Question 20:
Answer each of the following questions in one word or one sentence or as per exact requirement of the question:
Write the distance of a point P(a, b, c) from x-axis.
Answer:
We know that a general point (x, y, z) has distance from the x-axis.
∴ Distance of a point P(a, b, c) from x-axis =
Page No 27.25:
Question 21:
If a line makes angles 90° and 60° respectively with the positive directions of x and y axes, find the angle which it makes with the positive direction of z-axis.
Answer:
Let the direction cosines of the line be l, m and n.
We know that l2 + m2 + n2 = 1.
Let the line make angle θ with the positive direction of the z-axis.
cos2α+cos2β+cos2γ=1Here α=60 and β=45 and γ= θSo cos260+cos245+cos2θ=1cos2θ=1−14−12=14cosθ=±12So θ= 60 degree or 120.and here it is given that we have to find the angle made by negative z −axisSo cosθ=−12θ=120 degree
Page No 27.25:
Question 1:
For every point P (x, y, z) on the xy-plane,
(a) x = 0
(b) y = 0
(c) z = 0
(d) x = y = z = 0
Answer:
(c) z = 0
The Z-coordinate of every point on the XY-plane is zero.
Page No 27.25:
Question 2:
For every point P (x, y, z) on the x-axis (except the origin),
(a) x = 0, y = 0, z ≠ 0
(b) x = 0, z = 0, y ≠ 0
(c) y = 0, z = 0, x ≠ 0
(d) x = y = z = 0
Answer:
Page No 27.25:
Question 3:
A rectangular parallelopiped is formed by planes drawn through the points (5, 7, 9) and (2, 3, 7) parallel to the coordinate planes. The length of an edge of this rectangular parallelopiped is
(a) 2
(b) 3
(c) 4
(d) all of these
Answer:
Page No 27.25:
Question 4:
A parallelopiped is formed by planes drawn through the points (2, 3, 5) and (5, 9, 7), parallel to the coordinate planes. The length of a diagonal of the parallelopiped is
(a) 7
(b)
(c)
(d) none of these
Answer:
Page No 27.25:
Question 5:
The xy-plane divides the line joining the points (−1, 3, 4) and (2, −5, 6)
(a) internally in the ratio 2 : 3
(b) externally in the ratio 2 : 3
(c) internally in the ratio 3 : 2
(d) externally in the ratio 3 : 2
Answer:
Page No 27.25:
Question 6:
If the x-coordinate of a point P on the join of Q (2, 2, 1) and R (5, 1, −2) is 4, then its z-coordinate is
(a) 2
(b) 1
(c) −1
(d) −2
Answer:
Page No 27.25:
Question 7:
The distance of the point P (a, b, c) from the x-axis is
(a)
(b)
(c)
(d) none of these
Answer:
Page No 27.26:
Question 8:
Ratio in which the xy-plane divides the join of (1, 2, 3) and (4, 2, 1) is
(a) 3 : 1 internally
(b) 3 : 1 externally
(c) 1 : 2 internally
(d) 2 : 1 externally
Answer:
Page No 27.26:
Question 9:
If P (3, 2, −4), Q (5, 4, −6) and R (9, 8, −10) are collinear, then R divides PQ in the ratio
(a) 3 : 2 internally
(b) 3 : 2 externally
(c) 2 : 1 internally
(d) 2 : 1 externally
Answer:
Page No 27.26:
Question 10:
A (3, 2, 0), B (5, 3, 2) and C (−9, 6, −3) are the vertices of a triangle ABC. If the bisector of ∠ABC meets BC at D, then coordinates of D are
(a) (19/8, 57/16, 17/16)
(b) (−19/8, 57/16, 17/16)
(c) (19/8, −57/16, 17/16)
(d) none of these
Answer:
Disclaimer:This question is wrong, so the solution has not been provide.
Page No 27.26:
Question 11:
If O is the origin, OP = 3 with direction ratios proportional to −1, 2, −2 then the coordinates of P are
(a) (−1, 2, −2)
(b) (1, 2, 2)
(c) (−1/9, 2/9, −2/9)
(d) (3, 6, −9)
Answer:
Page No 27.26:
Question 12:
The angle between the two diagonals of a cube is
(a) 30°
(b) 45°
(c)
(d)
Answer:
(d)
Page No 27.26:
Question 13:
If a line makes angles α, β, γ, δ with four diagonals of a cube, then cos2 α + cos2 β + cos2 γ + cos2 δ is equal to
(a)
(b)
(c)
(d)
Answer:
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