Problem 10 - College Trigonometry

Question

Use the given pair of vectors \(\vec{v}\) and \(\vec{w}\) to find the following quantities. State whether the result is a vector or a scalar. $$ \begin{array}{lllll} \bullet \vec{v}+\vec{w} & \bullet \vec{w}-2 \vec{v} & \bullet\|\vec{v}+\vec{w}\| & \bullet\|\vec{v}\|+\|\vec{w}\| & \bullet\|\vec{v}\| \vec{w}-\|\vec{w}\| \vec{v} & \bullet\|\vec{w}\| \hat{v} \end{array} $$ Finally, verify that the vectors satisfy the Parallelogram Law $$\|\vec{v}\|^{2}+\|\vec{w}\|^{2}=\frac{1}{2}\left[\|\vec{v}+\vec{w}\|^{2}+\|\vec{v}-\vec{w}\|^{2}\right]$$ $$ \vec{v}=\frac{1}{2}(\hat{\imath}+\hat{\jmath}), \vec{w}=\frac{1}{2}(\hat{\imath}-\hat{\jmath}) $$

Step-by-Step Solution

Answer

The results are vectors or scalars; the Parallelogram Law is not satisfied.

Step 1: Calculate \( \vec{v} + \vec{w} \)

Substitute the given vector values for \( \vec{v} \) and \( \vec{w} \): \( \vec{v} = \frac{1}{2}(\hat{\imath} + \hat{\jmath}) \), \( \vec{w} = \frac{1}{2}(\hat{\imath} - \hat{\jmath}) \). Thus, \( \vec{v} + \vec{w} = \frac{1}{2}(\hat{\imath} + \hat{\jmath}) + \frac{1}{2}(\hat{\imath} - \hat{\jmath}) \). Simplifying, \( \vec{v} + \vec{w} = \hat{\imath} \). This result is a vector.

Step 2: Calculate \( \vec{w} - 2\vec{v} \)

First, calculate \( 2\vec{v} = 2(\frac{1}{2}(\hat{\imath} + \hat{\jmath})) = \hat{\imath} + \hat{\jmath} \). Next, for \( \vec{w} - 2\vec{v} = \frac{1}{2}(\hat{\imath} - \hat{\jmath}) - (\hat{\imath} + \hat{\jmath}) \), simplify to get \( \vec{w} - 2\vec{v} = -\frac{1}{2} \hat{\imath} - \frac{3}{2} \hat{\jmath} \). This result is a vector.

Step 3: Calculate \( \| \vec{v} + \vec{w} \| \)

Use the result from Step 1, \( \vec{v} + \vec{w} = \hat{\imath} \). The magnitude is \( \| \vec{v} + \vec{w} \| = \sqrt{(1)^2 + (0)^2} = 1 \). This result is a scalar.

Step 4: Calculate \( \| \vec{v} \| + \| \vec{w} \| \)

Calculate each magnitude separately: \( \| \vec{v} \| = \sqrt{(\frac{1}{2})^2 + (\frac{1}{2})^2} = \frac{1}{\sqrt{2}} \), and \( \| \vec{w} \| = \sqrt{(\frac{1}{2})^2 + (-\frac{1}{2})^2} = \frac{1}{\sqrt{2}} \). Summing them gives \( \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \sqrt{2} \). This result is a scalar.

Step 5: Calculate \( \|\vec{v}\| \vec{w} - \|\vec{w}\| \vec{v} \)

Knowing \( \|\vec{v}\| = \|\vec{w}\| = \frac{1}{\sqrt{2}} \), calculate the expression: \( \|\vec{v}\| \vec{w} - \|\vec{w}\| \vec{v} = \frac{1}{\sqrt{2}} \times \left(\frac{1}{2}(\hat{\imath} - \hat{\jmath})\right) - \frac{1}{\sqrt{2}} \times \left(\frac{1}{2}(\hat{\imath} + \hat{\jmath})\right) \). Simplifying yields \( -\frac{1}{\sqrt{2}}\hat{\jmath} \). This result is a vector.

Step 6: Calculate \( \|\vec{w}\| \hat{v} \)

\( \hat{v} \) is the unit vector in the direction of \( \vec{v} \), calculated as \( \hat{v} = \frac{\vec{v}}{\|\vec{v}\|} = (\frac{1}{2}, \frac{1}{2}) \times \sqrt{2} = (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})\hat{\imath} + (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})\hat{\jmath} \). Thus, \( \|\vec{w}\| \hat{v} = \frac{1}{\sqrt{2}} (\frac{1}{\sqrt{2}}\hat{\imath} + \frac{1}{\sqrt{2}}\hat{\jmath}) = \frac{1}{2}\hat{\imath} + \frac{1}{2}\hat{\jmath} \). This result is a vector.

Step 7: Verify Parallelogram Law

Calculate each term: \( \|\vec{v}\|^2 = \|\vec{w}\|^2 = (\frac{1}{\sqrt{2}})^2 = \frac{1}{2} \), \( \|\vec{v} + \vec{w}\|^2 = 1^2 = 1 \), and \( \|\vec{v} - \vec{w}\|^2 = \|\vec{0}\|^2 = 0 \). The left side becomes \( 1 \) and the right side becomes \( \frac{1}{2} \times (1 + 0) = \frac{1}{2} \). Therefore, the vectors do not satisfy the Parallelogram Law.

Key Concepts

Magnitude of a Vector

In vector mathematics, the magnitude is a number that describes the length or size of a vector in space. It is often thought of as the "distance" of the vector from the origin. For any vector \( \vec{v} = (v_1, v_2, v_3, ... , v_n) \), the magnitude \( \|\vec{v}\| \) is calculated using the Pythagorean theorem. For a 2D vector \( \vec{v} = (v_1, v_2) \), it is expressed as \( \|\vec{v}\| = \sqrt{v_1^2 + v_2^2} \).

When calculating the magnitude, the result is always a positive scalar value.

• Example: Given vector \( \vec{v} = \left(\frac{1}{2}, \frac{1}{2}\right) \), its magnitude is \( \|\vec{v}\| = \sqrt{\left(\frac{1}{2}\right)^2 + \left(\frac{1}{2}\right)^2} = \frac{1}{\sqrt{2}} \).
• This represents the distance from the origin to the point \( \left(\frac{1}{2}, \frac{1}{2}\right) \).

Calculating the magnitude helps us to understand how "long" or "strong" a vector is, which is crucial for various mathematical and physical applications.

Vector Addition

Vector addition is the process of combining two or more vectors to produce another vector. It follows the head-to-tail method where the end (head) of one vector is joined with the start (tail) of another. The sum of the vectors \( \vec{v} \) and \( \vec{w} \), represented as \( \vec{v} + \vec{w} \), results in a new vector.

For vectors in component form, they are added by summing their corresponding components:

• If \( \vec{v} = (v_1, v_2) \) and \( \vec{w} = (w_1, w_2) \), then \( \vec{v} + \vec{w} = (v_1 + w_1, v_2 + w_2) \).

For example, using the vectors \( \vec{v} = \frac{1}{2}(\hat{\imath} + \hat{\jmath}) \) and \( \vec{w} = \frac{1}{2}(\hat{\imath} - \hat{\jmath}) \), their sum is:
\( \vec{v} + \vec{w} = \frac{1}{2}(\hat{\imath} + \hat{\jmath}) + \frac{1}{2}(\hat{\imath} - \hat{\jmath}) = \hat{\imath} \).

In essence, vector addition is a straightforward linear operation that helps find resultant vectors for combinations of movements or forces in spatial dimensions.

Parallelogram Law

The Parallelogram Law is a fundamental principle in vector mathematics, stating that if two vectors are represented as two adjacent sides of a parallelogram, then their vector sum is represented by the diagonal of the parallelogram starting from the same point.

In mathematical terms, it involves verifying the equation:
\[ \|\vec{v}\|^2 + \|\vec{w}\|^2 = \frac{1}{2}\left[\|\vec{v} + \vec{w}\|^2 + \|\vec{v} - \vec{w}\|^2 \right] \]
For any two vectors \( \vec{v} \) and \( \vec{w} \), this equation is expected to hold true for their magnitudes.

• In some scenarios, such as with perpendicular vectors in special configurations, you might find deviations due to calculation or geometric specifics.

In the original problem, despite manually calculating specific values of magnitudes and verifying each term, the vectors seem not to satisfy this law. This may indicate a need for a reevaluation of calculations or understanding the vectors' geometric configurations. The Parallelogram Law is essential in physics and engineering for analyzing force systems and vector components.