2 Marks

Question 12 Marks

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

Answer

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

Question 22 Marks

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.

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Question 32 Marks

What are the direction cosines of Z-axis?

Answer

The z-axis makes angles 90°, 90° and 0° with x, y and z axes, respectively.
Therefore, the direction cosines of x-axis are cos 90°, cos 90°, cos 0°, i.e. 0, 0, 1.

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Question 42 Marks

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

Answer

If a line has direction ratios proportional to 2, -1 and -2, then its direction cosines are
$\frac{2}{\sqrt{(2)^2+(-1)+(-2)^2}},\frac{-1}{\sqrt{(2)^2+(-1)+(-2)^2}},\frac{-2}{\sqrt{(2)^2+(-1)+(-2)^2}}$
$=\frac{2}{3},-\frac{1}{3},-\frac{2}{3}$
Thus, the direction cosines are $=\frac{2}{3},-\frac{1}{3},-\frac{2}{3}$.

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Question 52 Marks

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

Answer

We know that if a line has direction ratio (a, b, c), then the cosine of its angle with the z-axis is given by
$\cos\gamma=\frac{\text{c}}{\sqrt{\text{a}^2+\text{b}^2+\text{c}^2}}$
Suppose the inclination of the line with direction ratio (0, 1, -1) with z-axis is $\gamma$.
Now,
$\cos\gamma=\frac{1}{\sqrt{0+1+1}}$
$=-\frac{1}{\sqrt{2}}$
Implies that $\lambda=\frac{3\pi}{4}$
Hence, the inclination of the line with z-axis is $\frac{3\pi}{4}$.

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Question 62 Marks

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

Answer

The distance of a general point (x, y, z) from x-axis is $\sqrt{\text{y}^2+\text{z}^2}$.
$\therefore$ Distance of the point (3, -5, 12) from x-axis $=\sqrt{(-5)^2+12^2}$
$=\sqrt{169}$
= $13\text{ units}$.

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Question 72 Marks

What are the direction cosines of X-axis?

Answer

The x-axis makes angles 0°, 90° and 90° with x, y and z axes, respectively.
Therefore, the direction cosines of x-axis are cos 90°, cos 0°, cos 90°, i.e. 1, 0, 0.

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Question 82 Marks

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

Answer

A line parallel to z-axis, makes an angle of 90°, 90° and 0° with the x, y, and z axes, respectively.
Thus, the direction cosines are given by
$\text{l}=\cos90^\circ=0$
$\text{m}=\cos90^\circ=0$
$\text{n}=\cos0^\circ=1$
Therefore, direction cosines of a line parallel to the z-axis 0, 0, 1.

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Question 92 Marks

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

Answer

A general point (x, y, z) is at a distance $\sqrt{\text{y}^2+\text{z}^2}$ of from the x-axis.
$\therefore$ Distance of the point (2, 3, 4) from x-axis
$=\sqrt{3^2+4^2}=\sqrt{25}$
$=5\text{ units}$.

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Question 102 Marks

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

Answer

The distance of a general point P (x, y, z) from XY-plane is z.
Thus, distance of (7, -2, 3) from XY-plane is 3.
Similarly, the distance of P (x, y, z) from YZ-plane is x.
Thus, distance of (7, -2, 3) from YZ-plane is 7.
The distance of P (x, y, z) from XZ-plane is y.
Thus, distance of (7, -2, 3) from XZ-plane is 2.

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Question 112 Marks

Define direction cosines of a direction line.

Answer

The direction cosines of a direction line segment are the cosines of the direction angles of the line segment. Let two points A(x₁, y₁, z₁) and (x₂, y₂, z₂) define the directed line segment AB.
The direction cosines of AB are given by cos
$\alpha=\frac{\text{x}_2-\text{x}_1}{\text{d}}$
$\cos\beta=\frac{\text{y}_2-\text{y}_1}{\text{d}}$
$\cos\gamma=\frac{\text{z}_2-\text{z}_1}{\text{d}}$
Here, d is the distance between A and B.

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Question 122 Marks

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

Answer

Suppose the line segment joining the points (a, b, c) and (-a, -c, -b) is divided by the XY-plane at a point R in the ratio $\lambda:1$.
Coordinates of R are
$\Big(\frac{\lambda(-\text{a})+1(\text{a})}{\lambda+1},\lambda(-\text{c})+\frac{1(\text{b})}{\lambda+1},\frac{\lambda(-\text{b})+1(\text{c})}{\lambda+1}\Big)$
Since R lies on XY-plane, Z-coordinate of R must be zero.
$\Rightarrow\frac{\lambda(-\text{b})+1(\text{c})}{\lambda+1}=0=\frac{\text{c}}{\text{b}}$
Thus, the required ratio is c/b : 1 or c : b.
Hence, the XY-plane divided the line in the ratio c : b.

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Question 132 Marks

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

Answer

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

Now,

$\text{l}=\frac{2}{\sqrt{2^2+(-1)^2+(-2)^2}}=\frac{2}{3}$

$\text{m}=\frac{-1}{\sqrt{2^2+(-1)^2+(-2)^2}}=\frac{-1}{3}$

$\text{n}=\frac{-2}{\sqrt{2^2+(-1)^2+(-2)^2}}=\frac{-2}{3}$

$\therefore$ The direction consines of the line are $\frac{2}{3},\frac{-1}{3},\frac{-2}{3}.$

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Question 142 Marks

If line makes angle $\alpha,\beta$ and $\gamma$ with the coordinate axes, find the value of $\cos2\alpha+\cos2\beta+\cos2\gamma$.

Answer

It is given that the line makes angles $\alpha,\beta,\gamma$ with the coordinate axis.
$\therefore\text{l}=\cos\alpha,\text{m}=\cos\beta$ and $\text{n}=\cos\gamma$
$\Rightarrow\text{l}^2+\text{m}^2+\text{n}^2=1$
$\Rightarrow\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1\ ......(1)$
Now,
$\cos2\alpha+\cos2\beta+\cos2\gamma$
$=(2\cos^2\alpha-1)+(2\cos^2\beta-1)+(2\cos^2\gamma-1)$
$=2\big(\cos^2\alpha+\cos^2\beta+\cos^2\gamma\big)-3$
$=2(1)-3$
$=-1$

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Question 152 Marks

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

Answer

Let the YZ-plane divides the line segment joining points P(-2, 5, 9) and Q(3, -2, 4) in the ratio k : 1.
Using the setion formula, the coordinates of the point of intersection are given by
$\Big(\frac{\text{k}(3)-2}{\text{k}+1},\frac{\text{k}(-2)+5}{\text{k}+1},\frac{\text{k}(4)+9}{\text{k}+1}\Big)$
On the YZ-plane, the X-coordinate of any point is zero.
$\frac{\text{k}(3)-2}{\text{k}+1}=0$
Implies that 3k - 2 = 0
Implies that $\text{k}=\frac{2}{3}$
Thus, the YZ-plane divides the line segment formed by joining the given points in the ratio 2 : 3 internally.

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Question 162 Marks

Answer the following quations 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 $\sqrt{\text{y}^2+\text{z}^2}$
$\therefore$ Distance of a point P(x, y, z) from x-axis $=\sqrt{\text{b}^2+\text{c}^2}$.

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Question 172 Marks

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 l, m and n be the direction cosines of a line.

$\text{l}=\cos90^{\circ}=0$

$\text{m}=\cos60^{\circ}=\frac{1}{2}$

$\text{n}=\cos30^{\circ}=\frac{\sqrt{3}}{2}$

$\therefore$ The direction consines of the line are $0,\frac{1}{2},\frac{\sqrt{3}}{2}.$

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Question 182 Marks

What are the direction cosines of Y-axis?

Answer

The y-axis makes angles 90°, 0° and 90° with x, y and z axes, respectively.
Therefore, the direction cosines of x-axis are cos 90°, cos 0°, cos 90°, i.e. 0, 1, 0.

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Question 192 Marks

If a line makes angles 90° and 60° respectively with the positive direction 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 $\theta$ with positive direction of the z-axis
$\alpha=90^\circ,\beta=60^\circ,\gamma=\theta$.
So, $\cos^290^\circ+\cos^260^\circ+\cos^2\theta=0$
$\Rightarrow0+\Big(\frac{1}{2}\Big)^2+\cos^2\theta=1$
$\Rightarrow\cos^2\theta=1-\frac{1}{4}$
$\Rightarrow\cos^2\theta=\frac{3}{4}$
$\Rightarrow\cos\theta=\pm\frac{\sqrt{3}}{4}$
$\Rightarrow\theta=30^\circ$or $150^\circ$

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Question 202 Marks

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

Answer

The coordinates of the projection of the point P(2, -3, 5) on Y-axis are (0, -3, 0) as both x and z coordinates of each point on the y-axis are equal to zero.

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