Important Questions For Class 12 Maths Chapter 11 Three Dimensional Geometry are available here to help the students who are appearing for the **CBSE board exams**. All the concepts of Three Dimensional Geometry for 12th standardÂ are important for students from the examination point of view to get more marks.

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## Important Questions & Answers for Class 12 Maths Chapter 11 Three Dimensional Geometry

**Q. 1: Find the direction cosines of the line passing through the two points (â€“ 2, 4, â€“ 5) and (1, 2, 3).**

**Solution:**

We know that the direction cosines of the line passing through two points P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) are given by

Using the distance formula,

\(PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\)From the given,

P(x_{1}, y_{1}, z_{1}) = (-2, 4, -5) and Q(x_{2}, y_{2}, z_{2}) = (1, 2, 3)

Hence, the direction cosines of the line joining the given two points are \(\frac{3}{\sqrt{77}}, \frac{-2}{\sqrt{77}}, \frac{8}{\sqrt{77}}\).

**Q. 2: Show that the points A (2, 3, â€“ 4), B (1, â€“ 2, 3) and C (3, 8, â€“ 11) are collinear.**

**Solution:**

We know that the direction ratios of the line passing through two points P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) are given by:

x_{2} – x_{1}, y_{2} – y_{1}, z_{2} – z_{1} or x_{1} – x_{2}, y_{1} – y_{2}, z_{1} – z_{2}

Given points are A (2, 3, â€“ 4), B (1, â€“ 2, 3) and C (3, 8, â€“ 11).

Direction ratios of the line joining A and B are:

1 â€“ 2, â€“ 2 â€“ 3, 3 + 4

i.e. â€“ 1, â€“ 5, 7.

The direction ratios of the line joining B and C are:

3 â€“1, 8 + 2, â€“ 11 â€“ 3

i.e., 2, 10, â€“ 14.

From the above, it is clear that direction ratios of AB and BC are proportional.

That means AB is parallel to BC. But point B is common to both AB and BC.

Hence, A, B, C are collinear points.

**Q. 3: If a line makes angles 90Â°, 135Â°, 45Â° with the x, y and z-axes respectively, find its direction cosines.**

**Solution:**

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

l = cos 90Â° = 0

m = cos 135Â° = -1/âˆš2

n = cos 45Â° = 1/âˆš2

Hence, the direction cosines of the line are 0, -1/âˆš2, and 1/âˆš2.

**Q. 4: Find the angle between the pair of lines given by **

**Solution:**

From the given,

Let Î¸ be the angle between the given pair of lines.

**Q. 5: Find the angle between the pair of lines given below.**

**(x + 3)/3 = (y -1)/5 = (z + 3)/4**

**(x + 1)/1 = (y – 4)/1 = (z – 5)/2**

**Solution:**

Given,

(x + 3)/3 = (y -1)/5 = (z + 3)/4

(x + 1)/1 = (y – 4)/1 = (z – 5)/2

The direction ratios of the first line are:

a_{1} = 3, b_{1} = 5, c_{1} = 4

The direction ratios of the second line are:

a_{2} = 1, b_{2} = 1, c_{2} = 2

Hence, the required angle is \(\theta=cos^{-1}(\frac{8\sqrt{3}}{15})\).

**Q. 6: Find the distance between the lines l _{1} and l_{2} given by:**

**Solution:**

Given two lines are parallel.

The distance between the two given lines is

**Q. 7: Show that the lines (x – 5)/7 = (y + 2)/-5 = z/1 and x/1 = y/2 = z/3 are perpendicular to each other.**

**Solution:**

Given lines are:

(x – 5)/7 = (y + 2)/-5 = z/1 and x/1 = y/2 = z/3

The direction ratios of the given lines are 7, -5, 1 and 1, 2, 3, respectively.

We know that,

Two lines with direction ratios a_{1}, b_{1}, c_{1} and a_{2}, b_{2}, c_{2} are perpendicular to each other if a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0

Therefore, 7(1) + (-5) (2) + 1 (3)

= 7 – 10 + 3

=0

Hence, the given lines are perpendicular to each other.

**Q. 8: Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector \(3\hat{i}+5\hat{j}-6\hat{k}.\)**

**Solution:**

Given, the normal vector is:

We know that the equation of the plane with position vector \(\vec{r}\) is given by \(\vec{r}.\hat{n}=d\).

Hence, the vector equation of the required plane is

**Q. 9: Find the intercepts cut off by the plane 2x + y â€“ z = 5.**

**Solution:**

Given plane is 2x + y – z = 5 â€¦â€¦(i)

Dividing both sides of the equation (i) by 5,

(â…–)x + (y/5) – (z/5) = 1

We know that,

The equation of a plane in intercept form is (x/a) + (y/b) + (z/c) = 1, where a, b, c are intercepts cut off by the plane at x, y, z-axes respectively.

For the given equation,

a = 5/2, b = 5, c = -5

Hence, the intercepts cut off by the plane are 5/2, 5 and -5.

**Q. 10: Find the equations of the planes that passes through three points (1, 1, 0), (1, 2, 1), and (â€“ 2, 2, â€“ 1).**

**Solution:**

Given points are (1, 1, 0), (1, 2, 1), and (â€“ 2, 2, â€“ 1).

Therefore, the plane will pass through the given three points.

We know that,

The equation of the plane through the points (x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2}) and (x_{3}, y_{3}, z_{3}) is

(x – 1)(-2) -(y – 1) (3) + z (3) = 0

-2x + 2 – 3y + 3 + 3z = 0

-2x – 3y + 3z + 5 = 0

-2x – 3y + 3z = -5

Therefore, 2x + 3y – 3z = 5 is the required Cartesian equation of the plane.

### Practice Questions For Class 12 Maths Chapter 11 Three Dimensional Geometry

- Find the direction cosines of a line whose direction ratios are 2, -6, 3. (Answer: 7)
- Find the direction cosines of a line that makes equal angles with the coordinate axes. (Answer: 1/âˆš3, 1/âˆš3, 1/âˆš3)
- Find the angles of triangle ABC whose vertices are A(-1, 3, 2), B (2, 3, 5) and C(3, 5, -2). (Answer: A = pi/2, B =cos-1(1/âˆš3), C =cos-1(âˆš2/5)
- Find the angles between the lines whose direction ratios are 3, 2, -6 and 1, 2, 2. ( Answer: cos-1(5/21)
- A line makes an angle 60 degree and 45 degrees with the positive direction of x-axis and y-axis respectively. What acute angle does it make with the z-axis? (Answer: 60 degrees)
- Show that the lines (x-1)/2=(y-2)/2=(z-3)/2 and (x-4)/5 = (y-1)/2 = z intersect each other. Also, find the point of intersection.
- Find the equation of the plane which is at a distance 3âˆš3 units from origin and the normal to which is equally inclined to coordinate axis.
- Find the angle between the lines whose direction cosines are given by the equation: l+m+n = 0, l
^{2}+m^{2}+n^{2}= 0. - O is the origin and A is (a,b,c). Find the direction cosines of the line OA and the equation of the plane through A at the right angle to OA.