Vectors and Vector Operations: Foundations for Multivariable Calculus

Vectors: Definition and Geometric Interpretation

What is a Vector?

Vectors are fundamental objects in mathematics and physics, representing quantities that possess both magnitude and direction. In multivariable calculus, vectors are used to describe points, directions, and physical quantities such as force and velocity.

• Flawed (but useful) Definition: A vector is a quantity with both magnitude and direction.

• Working Definition: A vector is an arrow in space, characterized by its tail (starting point) and head (ending point).

• Vector Coordinates: If a vector v is positioned with its tail at the origin and its head at the point (a, b, c), it is written as v = ⟨a, b, c⟩.

Vectors vs. Points

Vectors can be used to describe the displacement between two points in space. If Q:(a, b, c) and P:(d, e, f) are points, the vector PQ pointing from P to Q is:

• Formula: $PQ = \langle d-a, e-b, f-c \rangle$

• Example: The vector from P(1,2,3) to Q(4,6,8) is $PQ = \langle 4-1, 6-2, 8-3 \rangle = \langle 3, 4, 5 \rangle$

Vector Fields

In advanced calculus, vector fields are functions that assign a vector to every point in space. These are crucial in physics and engineering for modeling phenomena such as fluid flow and electromagnetic fields.

• Definition: A vector field is a function that maps each point in space to a vector.

• Example: The velocity field of a rotating fluid.

Vector Operations

Basic Operations in ℝ2 and ℝ3

Vectors can be added, subtracted, and multiplied by scalars. These operations are essential for manipulating vectors in calculus and physics.

• Vector Addition: $\mathbf{u} + \mathbf{v} = \langle u_1 + v_1, u_2 + v_2 \rangle$ (in ℝ2), $\mathbf{u} + \mathbf{v} = \langle u_1 + v_1, u_2 + v_2, u_3 + v_3 \rangle$ (in ℝ3)

• Vector Subtraction: $\mathbf{u} - \mathbf{v} = \langle u_1 - v_1, u_2 - v_2 \rangle$ (in ℝ2), $\mathbf{u} - \mathbf{v} = \langle u_1 - v_1, u_2 - v_2, u_3 - v_3 \rangle$ (in ℝ3)

• Scalar Multiplication: $c\mathbf{u} = \langle cu_1, cu_2 \rangle$ (in ℝ2), $c\mathbf{u} = \langle cu_1, cu_2, cu_3 \rangle$ (in ℝ3)

Magnitude (Length) of a Vector

The magnitude of a vector represents its length, calculated using the Pythagorean theorem.

• Magnitude in ℝ2: $|\mathbf{v}| = \sqrt{v_1^2 + v_2^2}$

• Magnitude in ℝ3: $|\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}$

• Example: For $\mathbf{u} = \langle -4, 0, -3 \rangle$, $|\mathbf{u}| = \sqrt{(-4)^2 + 0^2 + (-3)^2} = 5$

Dot Product and Orthogonality

Dot Product

The dot product is a fundamental operation that combines two vectors to produce a scalar. It is used to determine the angle between vectors and to test for orthogonality.

• Definition: $\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos \theta$

• Component Formula: $\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3$

• Orthogonality: Two vectors are orthogonal (perpendicular) if their dot product is zero.

• Example: $\mathbf{u} = \langle 1, -3, 2 \rangle$, $\mathbf{v} = \langle -2, 0, 1 \rangle$, $\mathbf{u} \cdot \mathbf{v} = 1 \times (-2) + (-3) \times 0 + 2 \times 1 = -2 + 0 + 2 = 0$

Orthogonal Projections

The orthogonal projection of one vector onto another represents the "shadow" that the first vector casts onto the second. This concept is essential in physics for calculating work and in geometry for decomposing vectors.

• Projection Formula: $\text{proj}_\mathbf{v} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v}$

• Scalar Component: $\text{scal}_\mathbf{v} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|}$

• Example: If two vectors are orthogonal, their projection is the zero vector.

Physical Interpretation: Work

In physics, the work done by a force is the dot product of the force vector and the displacement vector. Only the component of the force in the direction of motion contributes to the work.

• Work Formula: $W = |\mathbf{F}| |\mathbf{d}| \cos \theta = \mathbf{F} \cdot \mathbf{d}$

• Application: Calculating work in mechanics and engineering.

Cross Product and Areas

Cross Product

The cross product is an operation defined for vectors in three-dimensional space. It produces a vector that is orthogonal to both input vectors, with magnitude equal to the area of the parallelogram they span.

• Magnitude: $|\mathbf{u} \times \mathbf{v}| = |\mathbf{u}| |\mathbf{v}| \sin \theta$

• Direction: Determined by the right-hand rule; the resulting vector is perpendicular to both u and v.

• Component Formula: $\mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix}$

• Application: Used to compute torque, angular momentum, and area of parallelograms.

Summary Table: Vector Operations

Operation	Formula (ℝ3)	Result	Geometric Meaning
Addition	$\mathbf{u} + \mathbf{v} = \langle u_1 + v_1, u_2 + v_2, u_3 + v_3 \rangle$	Vector	Combines directions and magnitudes
Dot Product	$\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3$	Scalar	Measures alignment; zero if orthogonal
Cross Product	$\mathbf{u} \times \mathbf{v}$	Vector	Orthogonal to both; magnitude is area
Magnitude	$|\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}$	Scalar	Length of vector

Additional info: These notes cover foundational concepts for multivariable calculus, specifically vector operations, which are prerequisites for understanding lines, planes, surfaces, and vector fields in higher dimensions.