Question

Given a fixed vector \(\mathbf{v},\) there is an infinite set of vectors \(\mathbf{u}\) with the same value of proj\(_{\mathbf{v}} \mathbf{u}\). Find another vector that has the same projection onto \(\mathbf{v}=\langle 1,1,1\rangle\) as \(\mathbf{u}=\langle 1,2,3\rangle\).

Short answer

Question: Find another vector that has the same projection onto the vector \(\mathbf{v}=\langle 1,1,1\rangle\) as the given vector \(\mathbf{u}=\langle 1,2,3\rangle\). Answer: \(\mathbf{u'}=\langle 1,3,2\rangle\)

Step-by-step solution

Calculate the projection of \(\mathbf{u}\) onto \(\mathbf{v}\)

Using the projection formula, we can find the projection of \(\mathbf{u}\) onto \(\mathbf{v}\): proj\(_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}}\mathbf{v} = \frac{\langle 1,2,3\rangle \cdot \langle 1,1,1\rangle}{\langle 1,1,1\rangle \cdot \langle 1,1,1\rangle}\langle 1,1,1\rangle = \frac{6}{3}\langle 1,1,1\rangle = 2\langle 1,1,1\rangle\)

Find a vector orthogonal to \(\mathbf{v}\)

To find a vector orthogonal to \(\mathbf{v}\), we need a vector \(\mathbf{w}\) such that \(\mathbf{w} \cdot \mathbf{v} = 0\). We can choose a vector \(\mathbf{w}=\langle -1,1,0\rangle\), as this particular choice is orthogonal to \(\mathbf{v}\): \(\mathbf{w}\cdot\mathbf{v} = \langle -1,1,0\rangle \cdot \langle 1,1,1\rangle = (-1)(1)+1(1)+0(1)=0\)

Add the orthogonal vector to the projection to find \(\mathbf{u'}\)

Finally, we can add the orthogonal vector \(\mathbf{w}\) to the projection to find \(\mathbf{u'}\): \(\mathbf{u'} = \text{proj}_{\mathbf{v}} \mathbf{u} + \mathbf{w} = 2\langle 1,1,1\rangle + \langle -1,1,0\rangle = \langle 1,3,2\rangle\) So, another vector that has the same projection onto \(\mathbf{v}=\langle 1,1,1\rangle\) as \(\mathbf{u}=\langle 1,2,3\rangle\) is \(\mathbf{u'}=\langle 1,3,2\rangle\).

Key concepts

Orthogonal Vectors

Two vectors are orthogonal if they meet at right angles to each other, basically forming a 90-degree angle. This is one of the fundamental concepts in vector algebra and it simplifies understanding vector relationships. Mathematically, orthogonal vectors have a dot product of zero. So, if you have vectors \(\mathbf{a}\) and \(\mathbf{b}\), they are orthogonal if and only if \(\mathbf{a} \cdot \mathbf{b} = 0\).
For example, if you have vector \(\mathbf{v} = \langle 1, 1, 1 \rangle\) and you need a vector \(\mathbf{w}\) that is orthogonal to \(\mathbf{v}\), \(\mathbf{w} = \langle -1, 1, 0 \rangle\) works because the dot product \(\mathbf{w} \cdot \mathbf{v} = -1 \times 1 + 1 \times 1 + 0 \times 1 = 0\).

• Orthogonality is crucial for understanding vector spaces.
• Orthogonal vectors have no component in the same direction.
• This concept helps in decomposing vectors and simplifying problems.

Vector Addition

Vector addition involves combining two or more vectors to form a new vector. The process is simple: add the corresponding components of each vector.
For vectors \(\mathbf{a} = \langle a_1, a_2, a_3 \rangle\) and \(\mathbf{b} = \langle b_1, b_2, b_3 \rangle\), their sum \(\mathbf{a} + \mathbf{b} = \langle a_1 + b_1, a_2 + b_2, a_3 + b_3 \rangle\).
It's like putting two vectors head-to-tail and drawing a vector from the tail of the first to the head of the last.
This concept was used in the given problem, where the orthogonal vector \(\mathbf{w} = \langle -1, 1, 0 \rangle\) and the projection \(2 \langle 1, 1, 1 \rangle\) are added to get \(\mathbf{u'} = \langle 1, 3, 2 \rangle\).

• Vector addition is commutative: \(\mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}\).
• It is also associative: \((\mathbf{a} + \mathbf{b}) + \mathbf{c} = \mathbf{a} + (\mathbf{b} + \mathbf{c})\).
• Useful in physics for combining forces or velocities.

Dot Product

The dot product, also known as the scalar product, is a way to multiply two vectors to get a scalar value. It's used to find the angle between vectors and determine orthogonality, just to name a few applications.
The formula involves multiplying corresponding components of two vectors and then summing the results: \(\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\) for vectors \(\mathbf{a} = \langle a_1, a_2, a_3 \rangle\) and \(\mathbf{b} = \langle b_1, b_2, b_3 \rangle\).
In the exercise, the dot product is used to calculate the projection of \(\mathbf{u}\) onto \(\mathbf{v}\), demonstrating its utility in finding relationships between vectors.

• Indicates vector parallelism: if \(\mathbf{a} \cdot \mathbf{b} = 0\), they are orthogonal.
• Calculates the angle between vectors: \(cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\|\|\mathbf{b}\|}\).
• Serves as a basis for defining geometrical shapes like planes.