Question

(a) find the dot product v \(\cdot \mathbf{w} ;\) (b) find the angle between \(\mathbf{v}\) and \(\mathbf{w} ;\) (c) state whether the vectors are parallel, orthogonal, or neither. $$ \mathbf{v}=\mathbf{i}+\sqrt{3} \mathbf{j}, \quad \mathbf{w}=\mathbf{i}-\mathbf{j} $$

Short answer

Dot product: \( 1 - \sqrt{3} \). The angle is given by \( \cos \theta = \frac{1 - \sqrt{3}}{2 \sqrt{2}} \). Vectors are neither parallel nor orthogonal.

Step-by-step solution

Write down the given vectors

The vectors given are: \( \mathbf{v} = \mathbf{i} + \sqrt{3} \mathbf{j} \) and \( \mathbf{w} = \mathbf{i} - \mathbf{j} \).

Compute the dot product \( \mathbf{v} \cdot \mathbf{w} \)

To find the dot product, use the formula \( \mathbf{v} \cdot \mathbf{w} = v_1 w_1 + v_2 w_2 \). Here, \( \mathbf{v} = (1, \sqrt{3}) \) and \( \mathbf{w} = (1, -1) \). So, \( \mathbf{v} \cdot \mathbf{w} = 1 \cdot 1 + \sqrt{3} \cdot (-1) = 1 - \sqrt{3} \).

Use the dot product to find the angle

The formula for the angle \( \theta \) between two vectors is \( \cos \theta = \frac{\mathbf{v} \cdot \mathbf{w}}{\|\mathbf{v}\| \|\mathbf{w}\|} \). First, find the magnitudes: \( \|\mathbf{v}\| = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = 2 \) and \( \|\mathbf{w}\| = \sqrt{1^2 + (-1)^2} = \sqrt{2} \). Then substitute: \( \cos \theta = \frac{1 - \sqrt{3}}{2 \sqrt{2}} \).

Determine the type of vectors

Vectors are orthogonal if their dot product is zero. Here, \( \mathbf{v} \cdot \mathbf{w} = 1 - \sqrt{3} eq 0 \), thus the vectors are not orthogonal. They are parallel if \( \mathbf{v} \propto \mathbf{w} \), but \( (1, \sqrt{3}) eq k(1, -1) \), so the vectors are not parallel either. Therefore, the vectors are neither parallel nor orthogonal.

Key concepts

Vectors

Vectors are fundamental in both mathematics and physics. They are quantities that have both magnitude (length) and direction.
Think of an arrow: the length of the arrow represents magnitude, and the direction it points represents the direction.
Vectors are often represented in terms of their components along the x and y axes. For instance, the vector \(\mathbf{v} = \mathbf{i} + \sqrt{3} \mathbf{j}\) means it has an x-component of 1 and a y-component of \(\sqrt{3}\).
When working with vectors, you will often encounter operations like addition, subtraction, and the dot product.

Angle Between Vectors

The angle between two vectors is crucial for understanding their relationship. The formula to find the angle between two vectors \(\mathbf{v}\) and \(\mathbf{w}\) involves the dot product and their magnitudes:
\[\cos \theta = \frac{\mathbf{v} \cdot \mathbf{w}}{\|\mathbf{v}\| \|\mathbf{w}\| }\]
First, compute the dot product. In our exercise, the dot product is \( \mathbf{v} \cdot \mathbf{w} = 1 - \sqrt{3}\).
Next, find the magnitudes of the vectors. For \( \mathbf{v} \), the magnitude is \( 2 \), and for \( \mathbf{w} \), the magnitude is \( \sqrt{2} \).
Then substitute these values into the angle formula to get \( \cos \theta = \frac{1 - \sqrt{3}}{2\sqrt{2}} \).
Understanding the angle helps in analyzing whether vectors are pointing in the same, opposite, or perpendicular directions.

Orthogonal Vectors

Vectors are considered orthogonal (or perpendicular) if the angle between them is 90 degrees.
This is when their dot product is equal to zero.
The dot product of two vectors \( (a, b) \) and \( (c, d) \) is computed as \( ac + bd \).
If this sum is zero, then the vectors are orthogonal.
In our exercise, \( \mathbf{v} \cdot \mathbf{w} = 1 - \sqrt{3} \) which is not zero, so \( \mathbf{v} \) and \( \mathbf{w} \) are not orthogonal.
Orthogonal vectors are very important in various applications, including computer graphics and physics, mainly because they often simplify problems and solutions.

Parallel Vectors

Vectors are parallel if they point in the same or exact opposite direction. This means one of the vectors is a scalar multiple of the other.
For example, if \( \mathbf{v} = k \mathbf{w} \) for some scalar \( k \), then \( \mathbf{v} \) and \( \mathbf{w} \) are parallel.
In our exercise, comparing \( (1, \sqrt{3}) \) and \( (1, -1) \), there is no constant \( k \) that satisfies \( (1, \sqrt{3}) = k(1, -1) \).
Hence, \( \mathbf{v} \) and \( \mathbf{w} \) are not parallel.
Parallel vectors are often used in engineering and physics to simplify vector analysis and computations, as they ensure vectors maintain the same directional relationships.