Three Dimensional Geometry For Class 12

Three Dimensional Geometry for class 12 covers the important topics such as direction cosine and direction ratios of a line joining two points. Also, we will discuss here, the equation of lines and planes in space under different condition, the angle between line and plane, between two lines etc. You need to practice the questions to understand the topic better and based on the formulas as well.

To understand the different types of shapes and figures, this topic has been introduced in Maths. In real-world, almost all the objects are in a three-dimensional shape.  For example, there are many objects at home such as a table, chair, bed, kitchen utensils, etc. which have 3D geometry. In our primary classes, we have learned the basics of three-dimension geometry, but in 12th standard, we will learn the advanced version of it.

Direction Cosine and Direction Ratios of a line

Consider a line L passing through origin makes an angle of αβγ with x, y, and z-axes respectively, then the cosine of these angles is the direction cosine of the directed line L.

Any three numbers which are proportional to the direction cosines of a line are called the direction ratios of the line. Consider the direction cosine of line L be l, m, n and direction ratio a, b, c then a = λl, b = λm, c = λn, for non-zero λ ∈ R.

\(\frac{l}{a} = \frac{m}{b} = \frac{n}{c} = k\)

Then the direction cosine are:

\(l = \pm \frac{a}{a^{2} + b^{2} + c^{2}}\), \(m = \pm \frac{b}{a^{2} + b^{2} + c^{2}}\), \(n = \pm \frac{c}{a^{2} + b^{2} + c^{2}}\)

Note: If the given line in space does not pass through the origin, then, in order to find its direction cosines, we draw a line through the origin and parallel to the given line. Now take one of the directed lines from the origin and find its direction cosines as two parallel line have same set of direction cosines.

Important Formulas

Let us discuss some important formulas of three-dimensional geometry, based on which we will solve problems.

Relation between the direction cosines of a line

For a given line RS with direction cosines l, m, n, the relation between the direction cosines of a line RS is given by;

l2 + m2 + n2 = 1

Direction cosines of a line passing through two points

The direction cosines of the line segment joining the points P(x1, y1, z1) and Q(x2, y2, z2) are:

x2-x1/PQ, y2-y1/PQ, z2-z1/PQ

where PQ = √[(x2-x1)2+(y2-y1)2+(z2-z1)2]

Important Questions

Practise questions to prepare for this chapter and score good marks in the exam.

Question 1: If a line makes angles 90°, 135°, 45° 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.

Here let α = 90°, β = 135° and γ = 45°

As we know,

l = cos α, m = cos β and n = cos γ

Thus, direction cosines are:

l = cos 90° = 0

m = cos 135°= cos (180° – 45°) = -cos 45° = -1/√2

n = cos 45° = 1/√2

Therefore, the direction cosines of the line are 0, -1/√2, 1/√2.

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

Solution:

If the direction ratios of two lines segments are proportional, then the lines are collinear.

Let the given points are: A(2, 3, 4), B(−1, −2, 1) and C(5, 8, 7)

Direction ratio of the line joining the points A (2, 3, 4) and B (−1, −2, 1), are:

((−1−2), (−2−3), (1−4)) = (−3, −5, −3)

Where, a1 = -3, b1 = -5, c1 = -3

Direction ratio of the line joining the points B (−1, −2, 1) and C (5, 8, 7) are:

[(5− (−1)), (8− (−2)), (7−1)] = (6, 10, 6)

Where, a2 = 6, b2 = 10 and c2 =6

Since, it is clear that the direction ratios of AB and BC are of same proportions, hence,

a2/a1 = 6/-3 = -2

b2/b1 = 10/-5 = -2

c2/c1 = 6/-3 = -2

Therefore, the points A, B and C are collinear.

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