Question

Let \(\mathbf{v} \neq \mathbf{0}\) be a nonzero vector and let \(a \neq 0\) be a scalar. If \(\mathbf{u}\) is any vector, show that the projection of \(\mathbf{u}\) on \(\mathbf{v}\) equals the projection of \(\mathbf{u}\) on \(a \mathbf{v}\).

Short answer

The projections are equal: \( \text{proj}_{\mathbf{v}} \mathbf{u} = \text{proj}_{a\mathbf{v}} \mathbf{u} \).

Step-by-step solution

Define Projection Formula

The projection of vector \( \mathbf{u} \) onto vector \( \mathbf{v} \) is given by the formula: \[ \text{proj}_{\mathbf{v}} \mathbf{u} = \left( \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \right) \mathbf{v}. \] Similarly, the projection of \( \mathbf{u} \) onto \( a \mathbf{v} \) is: \[ \text{proj}_{a\mathbf{v}} \mathbf{u} = \left( \frac{\mathbf{u} \cdot (a\mathbf{v})}{(a\mathbf{v}) \cdot (a\mathbf{v})} \right) (a\mathbf{v}). \]

Simplify the Projection onto a\(a\mathbf{v} \)

Calculate the components of the projection formula for \( a\mathbf{v} \): 1. The dot product \( \mathbf{u} \cdot (a\mathbf{v}) = a(\mathbf{u} \cdot \mathbf{v}) \). 2. The dot product \( (a\mathbf{v}) \cdot (a\mathbf{v}) = a^2(\mathbf{v} \cdot \mathbf{v}) \). Substituting these into the projection formula: \[ \text{proj}_{a\mathbf{v}} \mathbf{u} = \left( \frac{a(\mathbf{u} \cdot \mathbf{v})}{a^2(\mathbf{v} \cdot \mathbf{v})} \right) (a\mathbf{v}). \]

Cancel and Simplify Terms

Simplify the expression for the projection onto \( a\mathbf{v} \): \[ \text{proj}_{a\mathbf{v}} \mathbf{u} = \left( \frac{\mathbf{u} \cdot \mathbf{v}}{a(\mathbf{v} \cdot \mathbf{v})} \right) a\mathbf{v}. \] The \( a \) terms cancel out:\[ = \left( \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \right) \mathbf{v}. \]

Conclude that Both Projections are Equal

The final simplified expression \( \left( \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \right) \mathbf{v} \) is identical to the projection of \( \mathbf{u} \) onto \( \mathbf{v} \), proving that \( \text{proj}_{\mathbf{v}} \mathbf{u} = \text{proj}_{a\mathbf{v}} \mathbf{u} \).

Key concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in vector algebra. It helps in finding the angle between two vectors and is crucial in computing vector projections.
To calculate the dot product between two vectors, you multiply corresponding components of each vector and then take the sum of these products. For example, given vectors \( \mathbf{a} = (a_1, a_2, a_3) \) and \( \mathbf{b} = (b_1, b_2, b_3) \), the dot product \( \mathbf{a} \cdot \mathbf{b} \) is given by:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \).

One of the essential properties of the dot product is that it is commutative, meaning \( \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a} \).
This operation not only provides a number that represents the degree of alignment of two vectors, but it also simplifies calculations like projections and finding orthogonality. The dot product is zero if vectors are perpendicular.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar, or a real number. This operation scales the vector without changing its direction, unless the scalar is negative, which reverses the direction.
Consider a vector \( \mathbf{v} = (v_1, v_2, v_3) \) and a scalar \( a \). The result of the scalar multiplication is:

• \( a\mathbf{v} = (av_1, av_2, av_3) \).

In our exercise, this concept helps demonstrate why the projection of a vector on \( a\mathbf{v} \) results similarly to the projection on \( \mathbf{v} \).
When projecting a vector onto another vector that is scaled by a scalar, the scaling affects both the length and direction, but when rationalizing through proportions, these factors cancel out, yielding the same projection. This is a crucial aspect that shows the consistency and reliability of vector operations.

Vector Spaces

Vector spaces are mathematical structures formed by a collection of vectors. These vectors adhere to specific rules of vector addition and scalar multiplication. Understanding vector spaces is essential to grasp more complex vector operations, like projections.
In a vector space, you can add two vectors, and the result will still be a vector within the same space. Similarly, when you multiply a vector by a scalar, the resulting vector remains in the same space.

• Properties include closure under addition and scalar multiplication, the existence of a zero vector, and the possibility of combining vectors and scalars to form linear combinations.

Working with projections, like in the exercise, involves using these principles to understand how vectors interact within a space. Projections show how one vector can be represented relative to another, and understanding the vector space ensures that these representations are consistent and meaningful, aiding intuition in various practical applications.