Vectors in \(\mathbb{R}^n\) and \(\mathbb{C}^n\)
Introduction to Vectors
A vector is a mathematical object that has both magnitude and direction. Vectors are essential in physics, engineering, and mathematics. They can be represented in different dimensions, such as real number space \(\mathbb{R}^n\) and complex number space \(\mathbb{C}^n\).
Visualization of Vectors in \(\mathbb{R}^3\)
Consider a three-dimensional space where vectors are represented as arrows from the origin to a point \((x, y, z)\). For example:
- Vector \( u = (2,3,4) \)
- Vector \( v = (-1,2.5,1) \)
- Vector \( w = (0.5,4,3) \)
Vector Operations with Examples
Let \( u = (2,4,-5) \) and \( v = (1,6,9) \). Then:
- \( u+v = (2+1, 4+6, -5+9) = (3,10,4) \)
- \( 7u = (14,28,-35) \)
- \( -v = (-1,-6,-9) \)
- \( 3u – 5v = (1,-18,-60) \)
Properties of Vectors
- \( (u+v)+w = u+(v+w) \)
- \( u+0 = u \)
- \( u+(-u) = 0 \)
- \( u+v = v+u \)
- \( k(u+v) = ku+kv \)
- \( (k+k’)u = ku+k’u \)
- \( (kk’)u = k(k’u) \)
- \( 1u = u \)
Dot (Inner) Product
The dot product of two vectors \( u \) and \( v \) in \( \mathbb{R}^n \) is given by:
\[ u \cdot v = a_1b_1 + a_2b_2 + … + a_n b_n \]
If \( u \cdot v = 0 \), then the vectors are orthogonal.
Example – Dot Product
- \( u = (1,2,3), v = (4,5,-1) \)
- \( u \cdot v = 1(4) + 2(5) + 3(-1) = 9 \)
- \( v \cdot w = 8 + 35 – 4 = 39 \)
Example – Orthogonal Vectors
Vectors \( u \) and \( v \) are orthogonal if their dot product is zero.
- \( u = (3, -2, 1), v = (2, 4, -8) \)
- \( u \cdot v = -10 \) (Not orthogonal)
- \( u = (2, 2, -2), v = (2, -1, 1) \)
- \( u \cdot v = 0 \) (Orthogonal)
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