Problem 23 - Algebra and Trigonometry

Question

Use the dot product to determine whether v and w are orthogonal. $$ \mathbf{v}=\mathbf{i}+\mathbf{j}, \quad \mathbf{w}=\mathbf{i}-\mathbf{j} $$

Step-by-Step Solution

Answer

Yes, the vectors \(\mathbf{v}=\mathbf{i}+\mathbf{j}\) and \(\mathbf{w}=\mathbf{i}-\mathbf{j}\) are orthogonal because their dot product is 0.

Step 1: Understand the definition of dot product

The dot product of two vectors \(\mathbf{a}\) and \(\mathbf{b}\) is defined as \(\mathbf{a}\cdot\mathbf{b}=|a||b|cos(θ)\) where |a| and |b| are magnitudes of \(\mathbf{a}\) and \(\mathbf{b}\) respectively and θ is the angle between the vectors. But for the vectors which are represented in terms of \(\mathbf{i}\) and \(\mathbf{j}\), the dot product \(\mathbf{a}\cdot\mathbf{b}\) = ai*bj + aj*bi.

Step 2: Calculate the dot product

\(\mathbf{v}\cdot\mathbf{w}\) = (1*1) + (1*-1) = 1 - 1 = 0.

Step 3: Check if dot product is zero

Since the dot product is equal to 0, \(\mathbf{v}\) and \(\mathbf{w}\) are orthogonal.

Key Concepts

Orthogonal Vectors

When two vectors are orthogonal, they are at right angles (90 degrees) to each other. In mathematical terms, this means that their dot product equals zero. The dot product is a way to multiply two vectors that results in a scalar value. It gives us an idea of how parallel or perpendicular the vectors are. If the dot product is zero, as it is in the case of vectors \( \mathbf{v} = \mathbf{i} + \mathbf{j} \) and \( \mathbf{w} = \mathbf{i} - \mathbf{j} \), it confirms that the vectors are orthogonal. This is because the contribution of each vector component cancels out, resulting in zero. Therefore, whenever asked to determine if two vectors are orthogonal, compute their dot product. A result of zero confirms the vectors are perpendicular.

Vector Magnitudes

The magnitude of a vector is a measure of its length. For a vector \( \mathbf{a} = a_i \mathbf{i} + a_j \mathbf{j} \), its magnitude is calculated using the formula \( |a| = \sqrt{a_i^2 + a_j^2} \). It is essentially the hypotenuse of a right triangle formed by the vector components. Understanding vector magnitudes is crucial when it comes to the dot product, as they are part of the formula \( \mathbf{a} \cdot \mathbf{b} = |a||b| \cos(\theta) \), which also relates to the angle between vectors.
When you know the vector magnitudes and the angle between them, you can easily calculate the dot product. Even though the magnitudes of \( \mathbf{v} \) and \( \mathbf{w} \) were not needed to determine orthogonality in this case, they are important for understanding other vector operations.

Angle Between Vectors

The angle between two vectors is determined with the help of the dot product formula \( \mathbf{a} \cdot \mathbf{b} = |a||b| \cos(\theta) \). This equation can be rearranged to \( \cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{|a||b|} \), allowing us to solve for the angle \( \theta \). When the dot product is zero, like between vectors \( \mathbf{v} \) and \( \mathbf{w} \), we know \( \cos(\theta) = 0 \) and hence \( \theta = 90^\circ \).
This angle correlates with orthogonality. Anytime two vectors are perpendicular, the angle between them is 90 degrees. By utilizing this relationship, you can use dot products to easily determine angles between vectors and assess their positional relationship.

Vector Components

Vector components break a vector down into its individual contributions in different directions, usually along the \( \mathbf{i} \) and \( \mathbf{j} \) axes for 2D vectors. For instance, in the vector \( \mathbf{v} = \mathbf{i} + \mathbf{j} \), the component along the \( \mathbf{i} \) axis is 1 and along the \( \mathbf{j} \) axis is also 1.
These components allow us to understand how much influence a vector has in a particular direction. When calculating the dot product, we multiply the corresponding components of different vectors. This is seen in our problem where the dot product \( \mathbf{v} \cdot \mathbf{w} \) is calculated as \( (1 \times 1) + (1 \times -1) = 0 \).
Understanding vector components helps demystify how vectors interact with each other, showing the detailed effect of each direction they contribute to their resultant orientation or interaction.