Vector Valued Function

In mathematics, a function usually gives one value for each input. But in many cases, one value is not enough to describe a situation. For example, position, velocity, and force have both size and direction. To represent such quantities, we use vector-valued functions. A vector-valued function gives a vector as its output instead of a single number. These functions help us describe motion and paths in space in a simple and clear way.

Vector Vector Valued Function can be written as:

Where

• t is the input variable (often representing time)
• f(t), g(t), h(t) are the component functions, each of which is a real-valued function.

Examples on Vector-Valued Functions

2D Vector-Valued Function:

This represents a parametric equation of a circle.

3D Vector-Valued Function:

This represents a space curve in three dimensions.

More generally, a vector-valued function is defined as:

where t is the independent variable and are scalar functions determining the components of the vector at each value of t.

Magnitude of Vectors:

The magnitude (or length) of a vector in n-dimensional space is given by:

For example, for in 2-dimensional space:

Unit Vector:

To find the unit vectorin the direction of a given vector , divide the vector by its magnitude:

Zero Vector:

The zero vector is a vector where all components are zero. In n-dimensional space:

The zero vector has a magnitude of 0 and does not have a specific direction.

Vector Operations:

Addition: The sum of two vectors and is

Subtraction: The difference between two vectors 𝑢 and 𝑣 is:

Scalar Multiplication: Multiplying a vector by a scalar gives:

Dot Product: The dot product of two vectors and is:

Cross Product: The cross product of two vectors and \vec{v} = \langle v_{1}, v_{2}, v_{3} \rangle in 3-dimensional space is:

Vector Components:

A vector in 𝑛-dimensional space can be broken down into its components along each axis. For a vector

• 𝑣_1​ is the component along the 𝑥-axis.
• 𝑣₂​ is the component along the 𝑦-axis.
• 𝑣_𝑛 is the component along the 𝑛-th axis.

These components represent the projections of the vector onto the respective axes. The original vector is the sum of these component vectors.

Practice Problems on Vector Valued Functions

Practice Problem 1 :Find the angle between the vectors v = (3, -4) and u = (5, 2).

Solution:

Step 1: Compute the dot product

Step 2: Compute the magnitudes

Step 3: Compute cos⁡(𝜃):

Step 4: Compute θ:

So, the angle between the vectors 𝑣v and 𝑢u is approximately 75.15^∘

Practice Problem 2: Let v = (-1, 3, -2) and u = (4, -2, 1). Compute:

a) The magnitude of u.

b) The unit vector in the direction of v.

c) The dot product of v and u.

d) The angle between v and u.

Solution:

a) The magnitude of 𝑢:

b) The unit vector in the direction of 𝑣:

c) The dot product of 𝑣 and 𝑢:

d) The angle between 𝑣 and 𝑢:

Practice Problem 4: Find the cross product of the vectors v = (2, 1, -3) and u = (-1, 4, 2).

Solution:

First component:

Second component:

Third component:

So, the cross product of the vectors 𝑣 = (2, 1, −3) and 𝑢 = (−1, 4, 2) is (14, −1, 9).

Practice Problem 5: Find a vector that is perpendicular to both v = (3, 1, -2) and u = (2, -1, 4).

Solution:

We compute

First component: 1⋅4 − (−2) ⋅ (−1) = 4 − 2 = 4 − 2 = 2

Second component: (−2)⋅2 − 3⋅4 = − 4 − 12 = − 4 − 12 = − 16

Third component: 3⋅(−1) − 1⋅ 2 = - 5

So, the vector (2,−16,−5) is perpendicular to both 𝑣 = (3, 1, −2) and 𝑢 = (2, −1, 4).