Problem 84 - Algebra and Trigonometry

Question

Prove that the projection of \(\mathbf{v}\) onto \(\mathbf{i}\) is \((\mathbf{v} \cdot \mathbf{i}) \mathbf{i}\)

Step-by-Step Solution

Answer

The formula for the projection of a vector \( \mathbf{v} \) onto another \( \mathbf{i} \) is indeed given by \( (\mathbf{v} \cdot \mathbf{i}) \mathbf{i} \) when \( \mathbf{i} \) is a unit vector.

Step 1: Define Projection

The projection of vector \( \mathbf{v} \) onto vector \( \mathbf{i} \) is given by the formula \( \text{projection}_{\mathbf{i}}\mathbf{v} = ((\mathbf{v} \cdot \mathbf{i}) / \lVert \mathbf{i} \rVert^2) \mathbf{i} \)

Step 2: Derive the Desired Formula

We aim to prove that the projection can also be written as \( (\mathbf{v} \cdot \mathbf{i}) \mathbf{i} \). If \( \mathbf{i} \) is a unit vector, meaning its length is 1, then \( \lVert \mathbf{i} \rVert^2 = 1 \), and the projection simplifies to the formula we want to prove.

Step 3: Prove the Formula

From the previous step, we have that for \( \mathbf{i} \) a unit vector, \( \text{projection}_{\mathbf{i}}\mathbf{v} = (\mathbf{v} \cdot \mathbf{i})\mathbf{i} \). Hence, the formula is correct in the case when \( \mathbf{i} \) is a unit vector. This constitutes a proof, because whenever we project onto a vector, we can always convert that vector into a unit vector by dividing by its length.

Key Concepts

Dot Product

The dot product is a fundamental operation in vector algebra. It is used to measure the similarity between two vectors regarding their direction.
When you calculate the dot product of two vectors, you multiply corresponding components and sum them up.
For vectors \( \mathbf{v} = \begin{bmatrix} v_1 & v_2 \end{bmatrix} \) and \( \mathbf{i} = \begin{bmatrix} i_1 & i_2 \end{bmatrix} \), the dot product, \( \mathbf{v} \cdot \mathbf{i} \), is given by:

• \( \mathbf{v} \cdot \mathbf{i} = v_1*i_1 + v_2*i_2 \)

This scalar quantity tells you how much of \( \mathbf{v} \) points in the direction of \( \mathbf{i} \).
Furthermore, if the dot product is zero, the vectors are perpendicular.

Unit Vector

A unit vector is a vector with a magnitude (or length) of 1.
It is essential for simplifying projections because projecting onto a unit vector is straightforward.
Symbolically, you often denote unit vectors with a hat symbol, like \( \mathbf{\hat{i}} \).

• To find a unit vector \( \mathbf{\hat{v}} \) in the direction of \( \mathbf{v} \), divide \( \mathbf{v} \) by its length: \( \mathbf{\hat{v}} = \frac{\mathbf{v}}{\|\mathbf{v}\|} \).

Using unit vectors in calculations like projection makes complex equations easier and helps convey direction without regard to magnitude.

Vector Length

The length of a vector, also known as its magnitude, denotes how long the vector is.
This is an important measure because it influences calculations like projection and normalization.
For a vector \( \mathbf{v} = \begin{bmatrix} v_1 & v_2 \text{,} \ldots \text{,} v_n \end{bmatrix} \), its length \( \|\mathbf{v}\| \) is calculated as:

• \( \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + \ldots + v_n^2} \)

Having a good grasp of vector length is critical, especially when transforming vectors into unit vectors or computing dot products.

Vector Proof

Vector proofs are logical arguments using geometric and algebraic properties of vectors to establish the validity of a statement.
In the context of the given exercise, the proof involves showing the equivalence between two different expressions for the projection of a vector.
Key steps in vector proofs typically include:

• Aligning all given expressions with known formulae like dot product or magnitude.
• Simplifying the equations using properties, such as the definition of a unit vector.
• Arriving at a consistent conclusion that meets the criteria of the problem statement.

The proof here demonstrates the power of unit vectors in simplifying projections and effectively ties vector concepts together, affirming the initial equation through logical steps.