RD Sharma XII Vol 2 2021 Solutions for Class 12 Science Maths Chapter 8 Direction Cosines And Direction Ratios are provided here with simple step-by-step explanations. These solutions for Direction Cosines And Direction Ratios are extremely popular among class 12 Science students for Maths Direction Cosines And Direction Ratios Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the RD Sharma XII Vol 2 2021 Book of class 12 Science Maths Chapter 8 are provided here for you for free. You will also love the ad-free experience on Meritnation’s RD Sharma XII Vol 2 2021 Solutions. All RD Sharma XII Vol 2 2021 Solutions for class 12 Science Maths are prepared by experts and are 100% accurate.
Page No 26.23:
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 26.23:
Question 2:
If a line has direction ratios 2, −1, −2, determine its direction cosines.
Answer:
Page No 26.23:
Question 3:
Find the direction cosines of the line passing through two points (−2, 4, −5) and (1, 2, 3).
Answer:
Page No 26.23:
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 26.23:
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:
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Page No 26.23:
Question 6:
Find the angle between the vectors with direction ratios proportional to 1, −2, 1 and 4, 3, 2.
Answer:
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Question 7:
Find the angle between the vectors whose direction cosines are proportional to 2, 3, −6 and 3, −4, 5.
Answer:
Page No 26.23:
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 26.23:
Question 9:
Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
Answer:
Page No 26.23:
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 26.23:
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 26.23:
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 26.23:
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 26.23:
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 26.23:
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 26.23:
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 26.24:
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 26.24:
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 26.24:
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 26.24:
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 26.25:
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 26.25:
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 26.25:
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 26.25:
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 26.25:
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 26.25:
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 26.25:
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 26.25:
Question 12:
The angle between the two diagonals of a cube is
(a) 30°
(b) 45°
(c)
(d)
Answer:
(d)
Page No 26.25:
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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Page No 26.25:
Question 14:
The coordinates of the foot of the perpendicular drawn from the point (2, 5, 7) on the x-axis are
(a) (2, 0, 0)
(b) (0, 5, 0)
(c) (0, 0, 7)
(d) (0, 5, 7)
Answer:
For given point (2, 5, 7)
Co-ordinates of foot of perpendicular an x-axis means y = 0, z = 0
i.e co-ordinates are given by (2, 0, 0)
Hence, the correct answer is option A.
Page No 26.25:
Question 15:
P is a point on the line segment joining the points (3, 2, –1) and (6, 2, –2). If x-coordinate of P is 5, then its y-coordinates is
(a) 2
(b) 1
(c) –1
(d) –2
Answer:
Line joining (3, 2, –1) and (6, 2, –2) has same y-coordinate
∴ Any point on the line joining (3, 2, –1) and (6, 2, –2) will have same y-coordinate i.e 2
Hence, the correct answer is option A.
Page No 26.25:
Question 16:
The distance of the point from y-axis is
Answer:
For any point (α, β, γ)
Let (0, β, 0) be a point on y-axis
∴ distance between (α, β, γ) and (0, β, 0) is
i.e
i.e
Hence, the correct answer is option D.
Page No 26.25:
Question 17:
The direction cosines of a line are k, k, k, then
(a) k > 0
(b) 0 < k < 1
(c) k = 1
(d)
Answer:
Given direction cosines of a line are k, k and k.
Say l = k, m = k, n = k
Since direction cosines are such that
l2 + m2 + n2 = 1
∴ k2 + k2 + k2 = 1
i.e 3k2 = 1
i.e k2 =
Hence, the correct answer is option D.
Page No 26.25:
Question 18:
The coordinates of the foot of the perpendicular drawn from the point (2, 3, 4) on the y - axis are
(a) (2, 3, 4)
(b) (2, 3, 4)
(c) (0, 3, 0)
(d) (2, 0, 4)
Answer:
Let the point A
Also, any point on is given by
Direction rations =
Since, AB is perpendicular to
Thus, the coordinates of foot of perpendicular is .
Page No 26.26:
Question 1:
The distance of the point (a, b, c) from y-axis is ____________.
Answer:
For given point (a, b, c)
Point perpendicular on y-axis is (0, b, 0)
∴ Distance of (a, b, c) from y axis is given by
i.e distance of (a, b, c) from y-axis is
Page No 26.26:
Question 2:
The distance of the point (a, b, c) from z-axis is ____________.
Answer:
For given point (a, b, c)
Point on z-axis will be (0, 0, c)
∴ Distance between (a, b, c) and z-axis, is given by
Hence, distance of (a, b, c) from z-axis is .
Page No 26.26:
Question 3:
If a line makes angles with x, y, z axes respectively, then its direction cosines are _____________.
Answer:
Since direction cosines are given by
l = cosα, m = cosβ and n = cosγ
where α is angle with x-axis, β is angle with y-axis and γ is angle with z-axis
Given, a line makes angles with x, y and z axes respectively.
Page No 26.26:
Question 4:
If a line makes angles α, β, γ with positive directions of the coordinate axes, then the value of cos 2α + cos 2β + cos 2γ is __________.
Answer:
Suppose a line makes angles α, β and γ with positive directions of the coordinate axes, then l = cosα, m = cosβ and n = cosγ
Here cos2α + cos2β + cos2γ
= 2cos2α – 1 + 2cos2β – 1 + 2cos2γ – 1
= 2(cos2α + cos2β + cos2γ) – 3
Since cos2α + cos2β + cos2γ
= l2 + m2 + n2
= 1
∴ cos2α + cos2β + cos2γ
= 2(cos2α + cos2β + cos2γ) – 3
= 2(1) – 3
i.e cos2α + cos2β + cos2γ = –1
Page No 26.26:
Question 5:
If a line angles α, β, γ with positive directions of the coordinate axes, then the value of sin2 α + sin2 β + sin2 γ is __________.
Answer:
Given, a line makes angles α, β and γ with positive directions of x-axis
Then l = cosα, m = cosβ and n = cosγ
Here sin2α + sin2β + sin2γ
= 1 – cos2α + 1 – cos2β + 1 – cos2γ
= 3 – (cos2α + cos2β + cos2γ)
= 3 – (l2 + m2 + n2)
= 3 – 1
i.e sin2α + sin2β + sin2γ = 2
Page No 26.26:
Question 6:
If a line makes an angle with each of y and z-axis, then the angle which it makes with x-axis, is _____________.
Answer:
Since a line makes an angle with y and z-axis
Then l = cos, m = cos, n = cos
Since l2 + m2 + n2 = 1
∴ The angle it makes with x-axis is .
Page No 26.26:
Question 7:
The direction cosines of the vector are ______________.
Answer:
Direction cosines of the vector is given by
i.e are the direction cosines.
Page No 26.26:
Question 8:
A unit vector making angle with x-axis, with y-axis and an acute angle with z-axis is ______________.
Answer:
Given, a unit vector makes angle with x-axis, with y-axis
i.e l = cos, m = cos, n = cos (say)
Since l2 + m2 + n2 = 1
Since γ is given to be acute
Page No 26.26:
Question 9:
If the projections of a line segment on the coordinates axes are 3, 4 and 5 then its length is equal to ____________.
Answer:
Since coordinates of a line segment from coordinate axes are 3, 4 and 5 then length is given by
Page No 26.26:
Question 10:
A vector of magnitude 21 having direction ratios proportional to 2, –3, 6 is ____________.
Answer:
Let us suppose vector is given by
Page No 26.26:
Question 11:
The direction cosines of the line joining points (4, 3, –5) and (–2, 1, –8) are ______________.
Answer:
For points given by (4, 3, –5) and (–2, 1, –8) line joining these points is
Hence direction cosines are given by
Page No 26.26:
Question 12:
If are direction cosines of a line, then the values of c are _______________.
Answer:
Let us suppose be the given direction cosines of a line
Page No 26.26:
Question 13:
If O is the origin and OP = 6 with direction ratios proportional to –1, 2, –2 then the coordinates of P are _____________.
Answer:
Let us suppose co-ordinate of P are (a, b, c)
Now, since direction ratios are proportional to –1, 2, –2.
i.e a = –λ, b = 2λ, c = –2λ
Since OP = 6
i.e λ2 + 4λ2 + 4λ2 = 36
i.e 9λ2 = 36
i.e λ = ±2 ∴ Co-ordinates of P are ( –2, 4, –4)
Page No 26.26:
Question 14:
The angle between the vectors with direction ratios proportional to 1, 1, 2 and is ________________.
Answer:
Let be a vector parallel to the vector having direction ratios 1, 1 and 2 and be the vector parallel to the vector having direction ratios
Page No 26.26:
Question 15:
If are the direction cosines of a line, then the values of n are ________________.
Answer:
Let us suppose, and n = n are the direction cosines of a line then l2 + m2 + n2 = 1
Page No 26.26:
Question 16:
If a line a makes angles with x, y and z axes respectively such that , then = _____________ .
Answer:
Given; a line makes angles α, β and γ with x, y and z axes respectively.
So, Let us suppose cosα, cosβ and cosγ represent direction cosines
Page No 26.26:
Question 17:
The total number of straight lines equally inclined with the coordinate axis is ____________.
Answer:
for any equally inclined line, α = β = γ (i.e direction cosines are same)
i.e cosα = cosβ = cosγ
i.e âcos2α + cos2β + cos2γ = 1
i.e 3 cos2α = 1
∴ Total 8 straight line segments are possible which are equally inclined to coordinate axes.
Page No 26.26:
Question 18:
zx-plane divides the line segment joining (2, 3, 1) and (6, 7, 1) in the ratio ______________.
Answer:
Given points are (2, 3, 1) and (6, 7, 1)
Since zx-plane divides the line segment joining (2, 3, 1) and (6, 7, 1) in the ratio say K:1
Sine y co-ordinate is 0 for zx-planei.e ratio is 3:7 externally.
Page No 26.27:
Question 1:
Define direction cosines of a directed line.
Answer:
Page No 26.27:
Question 2:
What are the direction cosines of X-axis?
Answer:
Page No 26.27:
Question 3:
What are the direction cosines of Y-axis?
Answer:
Page No 26.27:
Question 4:
What are the direction cosines of Z-axis?
Answer:
Page No 26.27:
Question 5:
Write the distances of the point (7, −2, 3) from XY, YZ and XZ-planes.
Answer:
Page No 26.27:
Question 6:
Write the distance of the point (3, −5, 12) from X-axis?
Answer:
Page No 26.27:
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 26.27:
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 26.27:
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 26.27:
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 26.27:
Question 11:
Write the inclination of a line with Z-axis, if its direction ratios are proportional to 0, 1, −1.
Answer:
Page No 26.27:
Question 12:
Write the angle between the lines whose direction ratios are proportional to 1, −2, 1 and 4, 3, 2.
Answer:
Page No 26.27:
Question 13:
Write the distance of the point P (x, y, z) from XOY plane.
Answer:
Page No 26.27:
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 26.27:
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 26.27:
Question 16:
Find the distance of the point (2, 3, 4) from the x-axis.
Answer:
Page No 26.27:
Question 17:
If a line has direction ratios proportional to 2, −1, −2, then what are its direction consines?
Answer:
Page No 26.27:
Question 18:
Write direction cosines of a line parallel to z-axis.
Answer:
Page No 26.27:
Question 19:
If a unit vector makes an angle and an acute angle θ with , then find the value of θ.
Answer:
Page No 26.27:
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 26.27:
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.
Page No 26.27:
Question 22:
If a line makes angles 90o, 135o, 45o with the x, y and z axes respectively, find its direction cosines.
Answer:
ans
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