Question

Vectors with equal projections Given a fixed vector \(\mathbf{v},\) there is an infinite set of vectors \(\mathbf{u}\) with the same value of \(\operatorname{proj}_{\mathbf{v}} \mathbf{u}\). Let \(\mathbf{v}=\langle 0,0,1\rangle .\) Give a description of all position vectors \(\mathbf{u}\) such that \(\operatorname{proj}_{\mathbf{v}} \mathbf{u}\) \(=\operatorname{proj}_{\mathbf{v}}\langle 1,2,3\rangle\).

Short answer

Question: Find all position vectors \(\mathbf{u}\) such that the projection of \(\mathbf{u}\) onto \(\mathbf{v}=\langle 0,0,1 \rangle\) is equal to the projection of \(\langle 1,2,3 \rangle\) onto \(\mathbf{v}\). Answer: All position vectors \(\mathbf{u}\) that satisfy the given condition have the form \(\mathbf{u} = \langle x, y, 3 \rangle\), where x and y can be any real numbers.

Step-by-step solution

Calculate the projection of \(\langle 1,2,3 \rangle\) onto \(\mathbf{v}\)

First, we need to calculate the projection of \(\langle 1,2,3 \rangle\) onto \(\mathbf{v}\). The formula for the projection of a vector \(\mathbf{a}\) onto \(\mathbf{b}\) is: \(\operatorname{proj}_\mathbf{b}\mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\lVert\mathbf{b}\rVert^2} \mathbf{b}\), where \(\mathbf{a} \cdot \mathbf{b}\) is the dot product of \(\mathbf{a}\) and \(\mathbf{b}\), and \(\lVert\mathbf{b}\rVert\) is the magnitude of \(\mathbf{b}\). Compute this for \(\mathbf{a}=\langle 1,2,3 \rangle\) and \(\mathbf{b}=\mathbf{v}=\langle 0,0,1 \rangle\).

Find the dot product and the magnitude

Since \(\mathbf{v}=\langle 0,0,1\rangle\), its magnitude \(\lVert\mathbf{v}\rVert\) is \(1\). Calculate the dot product of \(\langle 1,2,3 \rangle\) and \(\langle 0,0,1 \rangle\): \((1)(0)+(2)(0)+(3)(1)=3\).

Calculate the projection

Now that we have the dot product and the magnitude of \(\mathbf{v}\), we plug it into the formula for the projection: \(\operatorname{proj}_\mathbf{v}\langle 1,2,3 \rangle = \frac{3}{1^2} \langle 0,0,1 \rangle = \langle 0,0,3 \rangle\).

Find all position vectors \(\mathbf{u}\) such that \(\operatorname{proj}_\mathbf{v}\mathbf{u} = \langle 0,0,3 \rangle\)

We are looking for all vectors \(\mathbf{u}\) such that the projection of \(\mathbf{u}\) onto \(\mathbf{v}\) is equal to the projection of \(\langle 1,2,3 \rangle\) onto \(\mathbf{v}\). So we want to find all \(\mathbf{u}\) such that \(\operatorname{proj}_\mathbf{v}\mathbf{u} = \langle 0,0,3 \rangle\). We can rewrite this as \(\frac{\mathbf{u} \cdot \mathbf{v}}{\lVert \mathbf{v} \rVert^2} \mathbf{v} = \langle 0,0,3 \rangle\). To satisfy this equation, \(\mathbf{u}\) needs to have its z-component equal to 3, because the projection onto \(\mathbf{v}\) only cares about the z-component of \(\mathbf{u}\). The x and y components of \(\mathbf{u}\) can be any real numbers, as they do not affect the projection onto \(\mathbf{v}\). Therefore, this leads us to the following general description of all position vectors \(\mathbf{u}\): \(\mathbf{u} = \langle x, y, 3 \rangle\), where x and y can be any real numbers.

Key concepts

Dot Product

In vector mathematics, the dot product is a crucial operation that helps us understand the relationship between two vectors. The dot product of two vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \) is computed as \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \).

This operation results in a scalar (a single number rather than a vector) and reveals important information about alignment:

• If the dot product is positive, the vectors point in a somewhat similar direction.
• If it's zero, the vectors are perpendicular.
• If it's negative, they point in opposite directions.

In the context of projections, the dot product is used to ascertain the component of one vector along another. This is particularly useful when determining how one vector aligns with the basis vector in a specific direction, which was the focal point in this exercise.

Magnitude of Vectors

The magnitude of a vector is akin to measuring its length in space. For a vector \( \mathbf{v} = \langle a, b, c \rangle \), the magnitude equation is given by \( \lVert \mathbf{v} \rVert = \sqrt{a^2 + b^2 + c^2} \).

This mathematical expression provides insight into the size of the vector irrespective of its direction. In vector projections, as seen in the exercise, we utilize the magnitude in the denominator to normalize the direction of the vector we project onto.

In the example with vector \( \mathbf{v} = \langle 0,0,1 \rangle \), it simplifies beautifully because its magnitude is 1 \( (\sqrt{0^2 + 0^2 + 1^2} = 1) \).

• The straightforward nature of this magnitude (being 1) provides simplicity in calculations without altering the direction during projection.
• This characteristic is beneficial when performing projections in computations, as evident when solving the given problem.

Infinite Vector Solutions

Vectors can exhibit infinite solutions when certain conditions allow for variability along specific dimensions without affecting overall results. This concept becomes vivid in projection exercises, where one part of a vector doesn't influence the projection.

The given task illustrates this: once the projection goal is set \( \langle 0,0,3 \rangle \), vectors \( \mathbf{u} = \langle x, y, 3 \rangle \) offer infinite possibilities. Here, the x and y components can vary freely.

• This is due to orientation constraints where alterations in the non-significant components (x and y) do not influence the outcome once projected.
• The significant component (z in this case) is fixed to meet the projection criterion.

Understanding infinite solutions extends the flexibility in vector problems and aids in solving complex geometric issues where freedom in some vector dimensions supports the specific constraints required in others.