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 Class 12 Science Maths are prepared by experts and are 100% accurate.

#### 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.

Let the direction cosines of the line be l, m, n.

Now,

#### Question 2:

If a line has direction ratios 2, −1, −2, determine its direction cosines.

#### Question 3:

Find the direction cosines of the line passing through two points (−2, 4, −5) and (1, 2, 3).

#### Question 4:

Using direction ratios show that the points A (2, 3, −4), B (1, −2, 3) and C (3, 8, −11) are collinear.

#### 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). #### Question 6:

Find the angle between the vectors with direction ratios proportional to 1, −2, 1 and 4, 3, 2.

#### Question 7:

Find the angle between the vectors whose direction cosines are proportional to 2, 3, −6 and 3, −4, 5.

#### Question 8:

Find the acute angle between the lines whose direction ratios are proportional to 2 : 3 : 6 and 1 : 2 : 2.

#### Question 9:

Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.

#### 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).

#### 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).

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).

#### 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).

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).

#### Question 13:

Find the angle between the lines whose direction ratios are proportional to a, b, c and bc, ca, ab.

#### 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.

#### Question 15:

Find the direction cosines of the lines, connected by the relations: l + m +n = 0 and 2lm + 2lnmn = 0.

#### Question 16:

Find the angle between the lines whose direction cosines are given by the equations
(i) 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

(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 $l=-2m$, we have

$n=2×\left(-2m\right)+2m=-4m+2m=-2m$

When $l=-\frac{m}{2}$, we have

$n=2×\left(-\frac{m}{2}\right)+2m=-m+2m=m$

Thus, the direction ratios of two lines are proportional to

$-2m,m,-2m$ and $-\frac{m}{2},m,m$

Or $-2,1,-2$ and $-1,2,2$

So, vectors parallel to these lines are $\stackrel{\to }{a}=-2\stackrel{^}{i}+\stackrel{^}{j}-2\stackrel{^}{k}$ and $\stackrel{\to }{b}=-\stackrel{^}{i}+2\stackrel{^}{j}+2\stackrel{^}{k}$.

Let $\theta$ be the angle between these lines, then $\theta$ is also the angle between $\stackrel{\to }{a}$ and $\stackrel{\to }{b}$.

Thus, the angle between the two lines whose direction cosines are given by the given relations is $\frac{\mathrm{\pi }}{2}$.

#### 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

(c) z = 0

The Z-coordinate of every point on the XY-plane is zero.

#### 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

#### 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

#### 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) $\sqrt{38}$
(c) $\sqrt{155}$
(d) none of these

#### 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

#### 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

#### Question 7:

The distance of the point P (a, b, c) from the x-axis is
(a) $\sqrt{{b}^{2}+{c}^{2}}$

(b) $\sqrt{{a}^{2}+{c}^{2}}$

(c) $\sqrt{{a}^{2}+{b}^{2}}$

(d) none of these

#### 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

#### 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

#### 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

Disclaimer:This question is wrong, so the solution has not been provide.

#### 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)

#### Question 12:

The angle between the two diagonals of a cube is
(a) 30°

(b) 45°

(c) ${\mathrm{cos}}^{-1}\left(\frac{1}{\sqrt{3}}\right)$

(d) ${\mathrm{cos}}^{-1}\left(\frac{1}{3}\right)$

(d) ${\mathrm{cos}}^{-1}\left(\frac{1}{3}\right)$ #### 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) $\frac{1}{3}$

(b) $\frac{2}{3}$

(c) $\frac{4}{3}$

(d) $\frac{8}{3}$ #### 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)

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.

#### 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

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.

#### Question 16:

The distance of the point  from y-axis is

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  $\sqrt{{\left(\alpha -0\right)}^{2}+{\left(\beta -\beta \right)}^{2}+{\left(\gamma -0\right)}^{2}}\phantom{\rule{0ex}{0ex}}$

i.e $\sqrt{{\alpha }^{2}+{0}^{2}+{\gamma }^{2}}$
i.e $\sqrt{{\alpha }^{2}+{\gamma }^{2}}$

Hence, the correct answer is option D.

#### Question 17:

The direction cosines of a line are k, k, k, then
(a) > 0
(b) 0 < k < 1
(c) = 1
(d)

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$\frac{1}{3}$

Hence, the correct answer is option D.

#### 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)

Let the point A
Also, any point on $y-\mathrm{axis}$ is given by
$\therefore$Direction rations =
Since, AB is perpendicular to $y-\mathrm{axis}$
$⇒0\left(2\right)+\left(m\right)\left(-3-m\right)+0\left(4\right)=0\phantom{\rule{0ex}{0ex}}⇒m=-3$
Thus, the coordinates of foot of perpendicular is .

#### Question 1:

The distance of the point (a, b, c) from y-axis is ____________.

For given point (a, b, c
Point perpendicular on y-axis is (0, b, 0)
∴ Distance of (a, b, c) from axis is given by

i.e distance of (a, b, c) from y-axis is $\sqrt{{a}^{2}+{c}^{2}}$

#### Question 2:

The distance of the point (a, b, c) from z-axis is ____________.

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 $\sqrt{{a}^{2}+{b}^{2}}$.

#### Question 3:

If a line makes angles  with x, y, z axes respectively, then its direction cosines are _____________.

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.

#### 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 __________.

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

#### Question 5:

If a line angles α, β, γ with positive directions of the coordinate axes, then the value of sinα + sinβ + sin2 γ is __________.

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

#### Question 6:

If a line makes an angle with $\frac{\mathrm{\pi }}{4}$ each of y and z-axis, then the angle which it makes with x-axis, is _____________.

Since a line makes an angle $\frac{\mathrm{\pi }}{4}$ with y and z-axis
Then l = cos$\alpha$, m = cos$\frac{\mathrm{\pi }}{4}$, n = cos$\frac{\mathrm{\pi }}{4}$
Since l2 + m2 + n2 = 1

∴ The angle it makes with x-axis is $\frac{\mathrm{\pi }}{2}$.

#### Question 7:

The direction cosines of the vector $2\stackrel{^}{i}+2\stackrel{^}{j}-\stackrel{^}{k}$ are ______________.

Direction cosines of the vector $2\stackrel{^}{i}+2\stackrel{^}{j}-\stackrel{^}{k}$ is given by
i.e  are the direction cosines.

#### Question 8:

A unit vector making angle $\frac{\mathrm{\pi }}{4}$ with x-axis, $\frac{\mathrm{\pi }}{3}$ with y-axis and an acute angle with z-axis is ______________.

Given, a unit vector makes angle $\frac{\mathrm{\pi }}{4}$ with x-axis, $\frac{\mathrm{\pi }}{3}$ with y-axis
i.e l = cos$\frac{\mathrm{\pi }}{4}$, m = cos$\frac{\mathrm{\pi }}{3}$, n = cos$\gamma$ (say)
Since l2 + m2 + n2 = 1

Since γ is given to be acute

#### Question 9:

If the projections of a line segment on the coordinates axes are 3, 4 and 5 then its length is equal to ____________.

Since coordinates of a line segment from coordinate axes are 3, 4 and 5 then length is given by
$\sqrt{{\left(3-0\right)}^{2}+{\left(4-0\right)}^{2}+{\left(5-0\right)}^{2}}$
$=\sqrt{9+16+25}\phantom{\rule{0ex}{0ex}}=\sqrt{25+25}\phantom{\rule{0ex}{0ex}}=\sqrt{50}\phantom{\rule{0ex}{0ex}}=\sqrt{5×2×5}\phantom{\rule{0ex}{0ex}}$

#### Question 10:

A vector of magnitude 21 having direction ratios proportional to 2, –3, 6 is ____________.

Let us suppose vector is given by $\stackrel{\to }{r}=a\stackrel{^}{i}+b\stackrel{^}{j}+c\stackrel{^}{k}$

#### Question 11:

The direction cosines of the line joining points (4, 3, –5) and (–2, 1, –8) are ______________.

For points given by (4, 3, –5) and (–2, 1, –8) line joining these points  is $6\stackrel{^}{j}+2\stackrel{^}{j}+3\stackrel{^}{k}$

Hence direction cosines are given by $\frac{6}{7},\frac{2}{7},\frac{3}{7}$

#### Question 12:

If $\frac{1}{c},\frac{1}{c},\frac{1}{c}$ are direction cosines of a line, then the values of c are _______________.

Let us suppose  be the given direction cosines of a line

#### Question 13:

If O is the origin and OP = 6 with direction ratios proportional to –1, 2, –2 then the coordinates of P are _____________.

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)

#### Question 14:

The angle between the vectors with direction ratios proportional to 1, 1, 2 and  is ________________.

Let $\stackrel{\to }{a}$ be a vector parallel to the vector having direction ratios 1, 1 and 2  and $\stackrel{\to }{b}$ be the vector parallel to the vector having direction ratios

#### Question 15:

If  are the direction cosines of a line, then the values of n are ________________.

Let us suppose,  and n = n are the direction cosines of a line then l2 + m2 + n= 1

#### Question 16:

If a line a makes angles  with x, y and z axes respectively such that $\alpha +\beta =\frac{\mathrm{\pi }}{2}$, then $\gamma$ = _____________ .

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

#### Question 17:

The total number of straight lines equally inclined with the coordinate axis is ____________.

for any equally inclined line, α = β = γ (i.e direction cosines are same)
i.e cosα = cosβ = cosγ
i.e ​cos2α + cos2β + cos2γ = 1
i.e 3 cos2α = 1

∴ Total 8 straight line segments are possible which are equally inclined to coordinate axes.

#### Question 18:

zx-plane divides the line segment joining (2, 3, 1) and (6, 7, 1) in the ratio ______________.

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-plane

i.e ratio is 3:7 externally.

#### Question 1:

Define direction cosines of a directed line.

#### Question 2:

What are the direction cosines of X-axis?

#### Question 3:

What are the direction cosines of Y-axis?

#### Question 4:

What are the direction cosines of Z-axis?

#### Question 5:

Write the distances of the point (7, −2, 3) from XY, YZ and XZ-planes.

#### Question 6:

Write the distance of the point (3, −5, 12) from X-axis?

#### Question 7:

Write the ratio in which YZ-plane divides the segment joining P (−2, 5, 9) and Q (3, −2, 4).

#### 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.

#### 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.

#### Question 10:

Write the ratio in which the line segment joining (a, b, c) and (−a, −c, −b) is divided by the xy-plane.

#### Question 11:

Write the inclination of a line with Z-axis, if its direction ratios are proportional to 0, 1, −1.

#### Question 12:

Write the angle between the lines whose direction ratios are proportional to 1, −2, 1 and 4, 3, 2.

#### Question 13:

Write the distance of the point P (x, y, z) from XOY plane.

#### Question 14:

Write the coordinates of the projection of point P (x, y, z) on XOZ-plane.

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.

#### Question 15:

Write the coordinates of the projection of the point P (2, −3, 5) on Y-axis.

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.

#### Question 16:

Find the distance of the point (2, 3, 4) from the x-axis.

#### Question 17:

If a line has direction ratios proportional to 2, −1, −2, then what are its direction consines?

#### Question 18:

Write direction cosines of a line parallel to z-axis.

#### Question 19:

If a unit vector makes an angle and an acute angle θ with $\stackrel{^}{k}$, then find the value of θ.

#### 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.

We know that a general point (x, y, z) has distance $\sqrt{{y}^{2}+{z}^{2}}$ from the x-axis.

∴ Distance of a point P(a, b, c) from x-axis = $\sqrt{{b}^{2}+{c}^{2}}$

#### 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.

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=11412=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

#### Question 22:

If a line makes angles 90o, 135o, 45o with the x, y and z axes respectively, find its direction cosines.