1.1 Vector operations and properties

Vectors are the building blocks of multidimensional math. They help us describe motion, forces, and spatial relationships. In this section, we'll learn how to add, multiply, and manipulate vectors, which is crucial for understanding more complex concepts.

We'll dive into vector operations like addition and multiplication, as well as special products like dot and cross products. We'll also explore vector properties, including unit vectors and magnitude. These tools will be essential for solving real-world problems in physics and engineering.

Vector Operations

Adding and Multiplying Vectors

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• Vector addition combines two or more vectors to create a new vector (resultant vector)
  • Graphically, vector addition follows the parallelogram law or triangle method
  • Analytically, add the corresponding components of the vectors ($v_1 + v_2 = <v_{1x} + v_{2x}, v_{1y} + v_{2y}, v_{1z} + v_{2z}>$)
• Scalar multiplication modifies the magnitude of a vector without changing its direction
  • Multiply each component of the vector by the scalar value ($kv = <kv_x, kv_y, kv_z>$)
  • Negative scalar values reverse the direction of the vector

Dot Product and Cross Product

• Dot product (scalar product) calculates the scalar value resulting from the multiplication of two vectors
  • Formula: $v_1 \cdot v_2 = v_{1x}v_{2x} + v_{1y}v_{2y} + v_{1z}v_{2z}$
  • Geometrically, $v_1 \cdot v_2 = |v_1||v_2|\cos\theta$, where $\theta$ is the angle between the vectors
  • Dot product is commutative: $v_1 \cdot v_2 = v_2 \cdot v_1$
• Cross product (vector product) computes a new vector perpendicular to the plane containing the two input vectors
  • Formula: $v_1 \times v_2 = <v_{1y}v_{2z} - v_{1z}v_{2y}, v_{1z}v_{2x} - v_{1x}v_{2z}, v_{1x}v_{2y} - v_{1y}v_{2x}>$
  • Magnitude of the cross product: $|v_1 \times v_2| = |v_1||v_2|\sin\theta$
  • Direction determined by the right-hand rule
  • Cross product is not commutative: $v_1 \times v_2 = -v_2 \times v_1$

Vector Properties

Unit Vectors and Magnitude

• Unit vector represents the direction of a vector with a magnitude of 1
  • Denoted by a hat ($\hat{v}$) and calculated by dividing a vector by its magnitude: $\hat{v} = \frac{v}{|v|}$
  • Standard unit vectors: $\hat{i} = <1, 0, 0>$, $\hat{j} = <0, 1, 0>$, $\hat{k} = <0, 0, 1>$
• Vector magnitude (length) is the distance from the initial point to the terminal point of the vector
  • Calculated using the Euclidean norm: $|v| = \sqrt{v_x^2 + v_y^2 + v_z^2}$

Vector Relationships and Decomposition

• Vector projection calculates the component of one vector in the direction of another
  • Formula: $proj_{v_2}v_1 = \frac{v_1 \cdot v_2}{|v_2|^2}v_2$
  • Scalar projection: $comp_{v_2}v_1 = \frac{v_1 \cdot v_2}{|v_2|}$
• Orthogonal vectors are perpendicular to each other, with a dot product equal to zero ($v_1 \cdot v_2 = 0$)
• Vector decomposition breaks a vector into its components along specified directions
  • Rectangular decomposition: $v = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}$
  • Decomposition along non-standard directions using vector projection