Problem 17 - Algebra and Trigonometry

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. $$ \mathbf{v}=4 \mathbf{i}, \quad \mathbf{w}=\mathbf{j} $$

Step-by-Step Solution

Answer

(a) 0, (b) \(90^\circ \) or \( \frac{\pi}{2} \) radians, (c) orthogonal

Step 1: Find the dot product \( \textbf{v} \cdot \textbf{w} \)

The dot product of two vectors \(\mathbf{v} \) and \(\mathbf{w} \) is calculated as follows: \( \mathbf{v} \cdot \mathbf{w} = v_1 w_1 + v_2 w_2 \). Given \( \mathbf{v}=4 \mathbf{i} \) and \(\mathbf{w}=\mathbf{j} \), we identify the components: \( \mathbf{v} = (4,0) \) and \( \mathbf{w} = (0,1) \). Therefore, \( \mathbf{v} \cdot \mathbf{w} = 4 \cdot 0 + 0 \cdot 1 = 0 \).

Step 2: Find the magnitude of vectors \( \mathbf{v} \) and \( \mathbf{w} \)

The magnitude of a vector \( \mathbf{v} = (a,b) \) is given by \( \|\mathbf{v} \| = \sqrt{a^2 + b^2} \). For \( \mathbf{v} = (4,0) \), the magnitude is \( \|\mathbf{v} \| = \sqrt{4^2 + 0^2} = 4 \). For \( \mathbf{w} = (0,1) \), the magnitude is \( \|\mathbf{w} \| = \sqrt{0^2 + 1^2} = 1 \).

Step 3: Find the angle between \( \mathbf{v} \) and \( \mathbf{w} \)

The cosine of the angle \( \theta \) between two vectors can be found using the dot product formula: \( \cos(\theta) = \frac{\mathbf{v} \cdot \mathbf{w}}{\|\mathbf{v} \| \cdot \|\mathbf{w} \|} \). Substituting the values, \( \cos(\theta) = \frac{0}{4 \cdot 1} = 0 \). The angle \( \theta \) whose cosine is 0 is \( \theta = 90^\circ \) or \( \frac{\pi}{2} \) radians.

Step 4: Determine if vectors are parallel, orthogonal, or neither

Two vectors are parallel if they are scalar multiples of each other. They are orthogonal if their dot product is 0. Here, the dot product is 0, which means \( \mathbf{v} \) and \( \mathbf{w} \) are orthogonal. They are not parallel as \( \mathbf{v} \) and \( \mathbf{w} \) are not scalar multiples of each other.

Key Concepts

dot product

The dot product is a fundamental operation for vectors in algebra. It combines two vectors into a single number. Think of it as a way to measure how much one vector goes in the direction of another.
To calculate the dot product of two vectors \(\text{v} = (v_1, v_2)\) and \(\text{w} = (w_1, w_2)\), you follow this simple formula:
\[ \text{v} \cdot \text{w} = v_1 w_1 + v_2 w_2 \] In our example, with \(\text{v} = (4,0)\) and \(\text{w} = (0,1)\), you would perform:
\[\text{v} \cdot \text{w} = 4 \times 0 + 0 \times 1 = 0 \] This result tells us that these two vectors are indeed orthogonal.

magnitudes of vectors

The magnitude of a vector is like its length or size. It tells you how long the vector is, from the origin to its endpoint. The formula for the magnitude of a vector \(\text{v} = (a,b)\) is given by:
\[ \| \text{v} \| = \sqrt{a^2 + b^2} \] For our vectors:
For \(\text{v} = (4,0)\), the magnitude is:
\[ \| \text{v} \| = \sqrt{4^2 + 0^2} = 4 \]
For \(\text{w} = (0,1)\), the magnitude is:
\[ \| \text{w} \| = \sqrt{0^2 + 1^2} = 1 \]
Understanding magnitudes helps visualize and compare the sizes of vectors.

angle between vectors

Understanding the angle between two vectors helps in determining their directional relationship. You use the dot product to find this angle. The formula for the cosine of the angle (\theta) between two vectors \text{v} and \text{w} is:
\[ \cos(\theta) = \frac{\text{v} \cdot \text{w}}{\|\text{v}\| \|\text{w}\| } \] Substituting in our previous results:
\[ \cos(\theta) = \frac{0}{4 \times 1} = 0 \]
This corresponds to an angle of 90 degrees, or \(\frac{\text{π}}{2} \) radians.
Therefore, our vectors are perpendicular or orthogonal.

orthogonal vectors

Orthogonal vectors are a special pair of vectors that meet at a 90-degree angle, making them perpendicular to each other. One key indicator of orthogonality is the dot product; if the dot product of two vectors is zero, the vectors are orthogonal. In our example:
We found that:
\[ \text{v} \cdot \text{w} = 0 \]
This tells us that vectors \(\text{v} = 4 \text{i}\) and \(\text{w} = \text{j}\) are orthogonal. In summary, vectors are orthogonal when they are at right angles to each other and their dot product equals zero.