Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal to \(\mathbf{w},\) then \(\mathbf{u}+\mathbf{v}\) is orthogonal to \(\mathbf{w}\).

Short answer

The statement is true. If both \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal to \(\mathbf{w}\), then their sum \(\mathbf{u}+\mathbf{v}\) is also orthogonal to \(\mathbf{w}\).

Step-by-step solution

Understand orthogonality

Two vectors are orthogonal if their dot product equals to zero. This means \(\mathbf{u}\cdot \mathbf{w}=0\) if \(\mathbf{u}\) and \(\mathbf{w}\) are orthogonal. Similarly, \(\mathbf{v}\cdot \mathbf{w}=0\) if \(\mathbf{v}\) and \(\mathbf{w}\) are orthogonal.

Add the vectors

Sum the vectors \(\mathbf{u}\) and \(\mathbf{v}\) to get a new vector. This should give us the vector \(\mathbf{u}+\mathbf{v}\).

Explore the dot product

To check if \(\mathbf{u}+\mathbf{v}\) is orthogonal to \(\mathbf{w}\), we need to calculate their dot product. This can be done using the distribution law of dot product, that is \((\mathbf{u}+\mathbf{v})\cdot \mathbf{w} = \mathbf{u}\cdot \mathbf{w}+ \mathbf{v}\cdot \mathbf{w}\).

Substitute and verify

Remember that \(\mathbf{u}\cdot \mathbf{w}=0\) and \(\mathbf{v}\cdot \mathbf{w}=0\), substituting this in the equation \((\mathbf{u}+\mathbf{v})\cdot \mathbf{w} = \mathbf{u}\cdot \mathbf{w}+ \mathbf{v}\cdot \mathbf{w}\) will result in 0+0 which equals 0.

Key concepts

Vector Dot Product

Understanding the vector dot product is crucial when studying vectors and their interactions. Simply put, the dot product, also known as the scalar product, is an operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation is denoted by a 'dot' symbol, as in \( \mathbf{u} \cdot \mathbf{v} \).

When two vectors \( \mathbf{u} \) and \( \mathbf{v} \) are orthogonal, their dot product is zero, \( \mathbf{u} \cdot \mathbf{v} = 0 \). This zero result signifies that the vectors are at a 90-degree angle to each other. In the context of the exercise, if both \( \mathbf{u} \) and \( \mathbf{v} \) are orthogonal to a third vector \( \mathbf{w} \), it implies that \( \mathbf{u} \cdot \mathbf{w} = 0 \) and \( \mathbf{v} \cdot \mathbf{w} = 0 \).

Dot product calculations can be applied in various fields, such as physics to determine work done by a force or in computer graphics to calculate lighting on surfaces. It is a concept that extends beyond the realm of pure mathematics and plays a significant part in real-world applications.

Vector Addition

Vector addition is another fundamental concept when dealing with vectors. It enables us to combine vectors to form new ones. If we have two vectors, \( \mathbf{u} \) and \( \mathbf{v} \), vector addition is performed by adding the corresponding components of these vectors together. The resulting vector, \( \mathbf{u} + \mathbf{v} \), represents the combined effect of the original vectors.

For instance, if \( \mathbf{u} \) represents a force acting to the right and \( \mathbf{v} \) represents a force acting upwards, their sum \( \mathbf{u} + \mathbf{v} \) would represent the diagonal force that combines these two directions.

In our exercise, adding vectors \( \mathbf{u} \) and \( \mathbf{v} \) that are each orthogonal to \( \mathbf{w} \) will result in another vector that also maintains this orthogonality to \( \mathbf{w} \)—a fact that is verified by taking their dot product, as shown in other sections.

Distribution Law of Dot Product

The distribution law of dot product, or the distributive property, tells us how the dot product behaves with respect to vector addition. In mathematical terms, for any vectors \( \mathbf{u} \) , \( \mathbf{v} \) , and \( \mathbf{w} \), the distribution law can be expressed as: \[ (\mathbf{u} + \mathbf{v}) \cdot \mathbf{w} = \mathbf{u} \cdot \mathbf{w} + \mathbf{v} \cdot \mathbf{w} \].

This property makes it easier to compute the dot product of combined vectors, because you can calculate the dot products separately and then add the results.

Applying this to our original exercise, we can distribute \( \mathbf{w} \) across the sum of \( \mathbf{u} + \mathbf{v} \), simplifying the problem into calculating \( \mathbf{u} \cdot \mathbf{w} \) and \( \mathbf{v} \cdot \mathbf{w} \) individually. Since both of these products equal to zero, their sum also equals to zero. This confirms that \( \mathbf{u} + \mathbf{v} \) remains orthogonal to \( \mathbf{w} \)—a beautiful demonstration of how these key concepts interact in vector mathematics.