Vector in three dimensions | Part -2

Vectors in Three Dimensions: Operations and Properties

In three-dimensional geometry, a vector is defined by its components along the x, y, and z axes. A vector \( \vec{A} \) in 3D space is written as:

\[ \vec{A} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \]

In this context, \( \hat{i}, \hat{j}, \hat{k} \) are unit vectors pointing along the x, y, and z directions respectively, while \( a_1, a_2, a_3 \) are real values that serve as the components of the vector.

1. Vector Addition

Given two vectors: \[ \vec{A} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}, \quad \vec{B} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \] their sum is: \[ \vec{A} + \vec{B} = (a_1 + b_1) \hat{i} + (a_2 + b_2) \hat{j} + (a_3 + b_3) \hat{k} \]

2. Vector Subtraction

Subtracting one vector from another results in: \[ \vec{A} - \vec{B} = (a_1 - b_1) \hat{i} + (a_2 - b_2) \hat{j} + (a_3 - b_3) \hat{k} \]

3. Scalar Multiplication

Given a scalar ( k ), the following properties apply: \[ k \vec{A} = k a_1 \hat{i} + k a_2 \hat{j} + k a_3 \hat{k} \] This operation stretches or shrinks the vector.

4. Equality of Two Vectors

Vectors \( \vec{A} \) and \( \vec{B} \) are considered equal only when each of their corresponding components matches exactly: \[ a_1 = b_1, \quad a_2 = b_2, \quad a_3 = b_3 \]

5. Direction Ratios (DRs)

The direction ratios of a vector \( \vec{A} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \) are the coefficients \( a_1 : a_2 : a_3 \). These ratios show the vector's inclination with respect to the axes.

6. Direction Cosines (DCs)

The direction cosines are the cosines of angles \( \alpha, \beta, \gamma \) made by the vector with the x, y, and z axes, respectively: \[ \cos \alpha = \frac{a_1}{|\vec{A}|}, \quad \cos \beta = \frac{a_2}{|\vec{A}|}, \quad \cos \gamma = \frac{a_3}{|\vec{A}|} \] where: \[ |\vec{A}| = \sqrt{a_1^2 + a_2^2 + a_3^2} \]

The identity: \[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]. This holds true for any vector that is non-zero.

7. Resultant of Two Vectors

The resultant of vectors \( \vec{A} \) and \( \vec{B} \) is: \[ \vec{R} = \vec{A} + \vec{B} \] The direction cosines of \( \vec{R} \) can also be found using its components.

Conclusion

Grasping vector operations in three-dimensional space is essential for tackling challenges in physics and engineering fields. Vector addition, subtraction, scalar multiplication, direction ratios, and direction cosines allow us to analyze forces, motion, and positions in space.