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Every student has studied the concept of vectors in the middle school itself. However, it was usually related to the concepts of physics. By definition, a vector is defined as the quantity which has both magnitudes as well as direction. Some examples of vector quantities are velocity, displacement, acceleration, weight, momentum, force, etc.

In maths, the students are taught concepts of vector algebra and their operations, which help them solve questions related to their values and the other concepts. The key topics covered in the Vector Algebra chapter of Class 12 include:

Basic concepts.

Types of vectors.

Addition of vectores.

Direction cosines and ratios.

Consider a three-dimensional plane with origin O(0, 0, 0) and a point P located on the plane with the coordinates (x, y, z). The position vector is given by:

\[OP = \sqrt{x^{2} + y^{2} + z^{2}}\]

Relationship of magnitude, direction cosines, and direction ratios.

Let the magnitude be r.

The direction ratios have the coordinates (a, b, c).

The direction cosines have the coordinates (l, m, n).

The relation among the direction cosines and direction ratios is as follows:

l = a/r, m = b/r, n = c/r.

The various types of vectors included in the Class 12 concepts are as follows:

Null Vector or Zero Vector

The magnitude of these vectors is 0, and their initial and final points coincide. They are denoted by 0(vector). It does not have a definite direction. The zero vector is represented by AA(vector), BB(vector).

Unit Vector

A vector that has a unit length is called a unit vector.

Collinear Vectors

These are defined as those vectors parallel to the same line, irrespective of the direction and magnitude they hold.

Coinitial Vectors

The vectors which have the same initial point are called coinitial vectors.

Equal Vectors

Vectors having the same magnitude and same direction are said to be equal irrespective of their initial positions.

Negative Vectors

The vectors which are equal in magnitude but have opposite directions are called negative vectors.

Direction Cosines

Once you are well acquainted with the formula of position vectors, you must know the meaning of direction coines and direction cosines formula to solve the problems.

In the given figure, the position vector OP makes positive angles, namely, ⍺, 𝛽, and 𝛾 with the x-axis, y-axis and z-axis, respectively. The angles so formed are called direction angles. The direction cosines of a vector can be found out by taking the cosines of the above-mentioned angles.

Therefore, the direction cosines in a plane are given by the formulae:

cos ⍺ = x/r(vector)

cos 𝛽 = y/r(vector)

cos 𝛾 = z/r(vector)

We can rewrite the above equations in the form:

\[cos \alpha = \frac{x}{\sqrt{x^{2} + y^{2} + z^{2}}}\]

\[cos \beta = \frac{y}{\sqrt{x^{2} + y^{2} + z^{2}}}\]

\[cos \gamma = \frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}}\]

Direction Ratios

The direction ratios of a vector are given as a product of magnitude and cosines of the given vector. These are given by the direction cosine formula of:

a = lr

b = mr

c = nr

The above mentioned direction ratios are proportional to the direction cosines. These can be expressed by the formulae:

\[l/a = \frac{1}{\sqrt{x^{2} + y^{2} + z^{2}}}\]

\[m/b = \frac{1}{\sqrt{x^{2} + y^{2} + z^{2}}}\]

\[n/c = \frac{1}{\sqrt{x^{2} + y^{2} + z^{2}}}\]

At Vedantu, we provide you with comprehensive notes of direction ratios and cosines, allowing you to revise the concepts quickly and easily. Moreover, we provide you with the option of downloading these notes in PDF format, making the study process seamless. So, if you are preparing for your board or competitive examinations, you should be thorough with these concepts. Alongside, vector algebra forms an integral part of coordinate geometry in mathematics. This makes it quintessential for students to learn and understand the formulae and concepts well.

FAQ (Frequently Asked Questions)

Q1. Consider that a Given Line Makes Angles of 90°, 135°, and 45° Respectively with the X-Axis, Y-Axis, and Z-Axis. What are its Direction Cosines?

Ans. The direction cosines are given by:

l = cos 90° = 0

m = cos 135° = - 1/√2

n = cos 45° = 1/√2

Q2. What are the Direction Cosines of a Line Which is Aligned in Equal Angles with the Coordinate Axes?

Ans. Let the angle made by the line with each of the coordinate axes be 𝛼.

Then, l = cos 𝛼, m = cos 𝛼, n = cos 𝛼

Following this, l^{2} + m^{2} + n^{2} = 1.

Therefore,

𝛼^{2} + 𝛼^{2} + 𝛼^{2} = 1

cos 𝛼 = 1/√3

Thus, the direction cosines so formed are cos 𝛼 = 1/√3.

Q3. Consider that a Given Point P has Coordinates (3, 4, 5). Find Out the Direction Cosines and Direction Ratios of the Given Point and Take O(0, 0, 0) as its Reference.

Ans. Represent the given point on a cartesian plane.

Then, according to the formulae:

cos ⍺ = x/√(x^{2} + y^{2} + z^{2})

cos 𝛽 = y/√(x^{2 }+ y^{2 }+ z^{2})

cos 𝛾 = z/√(x^{2 }+ y^{2 }+ z^{2})

Here,

√(x^{2} + y^{2} + z^{2}) = √(9 + 16 + 25)

= √50 = 5√2

Therefore, by applying the formulae:

l = 3/5√2

m = 4/5√2

n = 1/√2

So, the direction ratios will be in the ratio of 3:4:5.

Therefore,

a = 3

b = 4

c = 5