Question

If \(\vec{T}(t)\) is a unit tangent vector, what is \(\|\vec{T}(t)\| ?\)

Short answer

The magnitude of \( \vec{T}(t) \) is 1.

Step-by-step solution

Understand the Unit Tangent Vector

A unit tangent vector \( \vec{T}(t) \) is a vector that is tangent to the curve at a point and has a magnitude (or length) of 1. It is called a unit vector because its length is 1.

Recall the Definition of Magnitude

The magnitude, or norm, of a vector \( \vec{v} \) is denoted by \( \| \vec{v} \| \) and is calculated using the Pythagorean theorem in the case of a 2D or 3D vector. It represents the 'length' of the vector.

Apply the Unit Tangent Vector Property

Since \( \vec{T}(t) \) is a unit tangent vector, by definition, its magnitude is equal to 1. Therefore, the norm \( \| \vec{T}(t) \| = 1 \) regardless of any specific values of \( t \).

Key concepts

Vector Magnitude

The magnitude of a vector tells us how "long" the vector is. Just like how we measure the length of a string with a ruler, the magnitude gives us the size of a vector as a single scalar number. If you imagine a vector as an arrow pointing in space, the magnitude is the length of that arrow.

To calculate the magnitude of a vector in a two or three-dimensional space, we use the Pythagorean theorem. The formula for the magnitude of a vector \( \vec{v} = \langle v_1, v_2, v_3 \rangle \) is:

• In 2D: \( \|\vec{v}\| = \sqrt{v_1^2 + v_2^2} \)
• In 3D: \( \|\vec{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2} \)

This use of the Pythagorean theorem helps us understand how the components of the vector combine to give its overall length.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In mathematical terms, if a triangle has sides of length \( a \) and \( b \) with \( c \) as the hypotenuse, then:

• \( a^2 + b^2 = c^2 \)

This principle is not only a cornerstone in geometry but also underpins the calculation of the magnitude of vectors in both two and three dimensions.

For vectors, we consider the vector as forming a right triangle with its components as the legs. This allows the Pythagorean theorem to be used as mentioned before to find the magnitude, or the "norm," of the vector.

Norm of a Vector

The norm of a vector, often referred to as its magnitude, is a measure of the vector's size or length. You can think of it as how far the vector reaches from its initial point toward its terminal point. The norm is calculated similarly to the magnitude using the Pythagorean theorem.

For any vector \( \vec{v} = \langle v_1, v_2, v_3 \rangle \), the norm \( \| \vec{v} \| \) is given by:

• \( \|\vec{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2} \) in a three-dimensional space.
• In a two-dimensional space, the formula simplifies to \( \|\vec{v}\| = \sqrt{v_1^2 + v_2^2} \).

The norm of a vector is always non-negative, and it reaches zero only when the vector itself is a zero vector (a point at the origin with no length).

Understanding the norm is crucial, particularly in problems involving unit vectors. A unit vector, by definition, already has a norm of 1. This means it maintains its direction without affecting its length, making it an essential concept in physics and engineering when dealing with directions and trajectories.