RD Sharma XII Vol 2 2018 Solutions for Class 12 Science Math 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 Math 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 2018 Book of class 12 Science Math 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 2018 Solutions. All RD Sharma XII Vol 2 2018 Solutions for class 12 Science Math 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:

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 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) $\frac{8}{3}$

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