Problem 9 - 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 $$\|\vec{v}\|^{2}+\|\vec{w}\|^{2}=\frac{1}{2}\left[\|\vec{v}+\vec{w}\|^{2}+\|\vec{v}-\vec{w}\|^{2}\right]$$ $$ \vec{v}=3 \hat{\imath}+4 \hat{\jmath}, \vec{w}=-2 \hat{\jmath} $$

Step-by-Step Solution

Answer

The parallelogram law holds true, confirming the calculations.

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

To find \( \vec{v}+\vec{w} \), simply add the corresponding components of each vector. \( \vec{v} = 3\hat{\imath} + 4\hat{\jmath} \) and \( \vec{w} = -2\hat{\jmath} \).\[ \vec{v}+\vec{w} = (3\hat{\imath} + 4\hat{\jmath}) + (0\hat{\imath} - 2\hat{\jmath}) = 3\hat{\imath} + 2\hat{\jmath} \]. The result is a vector.

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

Multiply \( \vec{v} \) by 2: \( 2\vec{v} = 2(3\hat{\imath} + 4\hat{\jmath}) = 6\hat{\imath} + 8\hat{\jmath} \). Subtract \( 2\vec{v} \) from \( \vec{w} \): \[ \vec{w}-2\vec{v} = (-2\hat{\jmath}) - (6\hat{\imath} + 8\hat{\jmath}) = -6\hat{\imath} -10\hat{\jmath} \]. This also results in a vector.

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

First, find \( \vec{v}+\vec{w} = 3\hat{\imath} + 2\hat{\jmath} \) from Step 1. Compute the magnitude: \[ \|3\hat{\imath} + 2\hat{\jmath}\| = \sqrt{3^2 + 2^2} = \sqrt{13} \]. This result is a scalar.

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

First, find the magnitudes: \( \|\vec{v}\|=\sqrt{3^2 + 4^2} = \sqrt{25} = 5 \) and \( \|\vec{w}\|=\sqrt{(-2)^2} = 2 \). Sum these magnitudes: \[ \|\vec{v}\|+\|\vec{w}\| = 5 + 2 = 7 \]. This result is a scalar.

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

Using \( \|\vec{v}\|=5 \) and \( \|\vec{w}\|=2 \), calculate each term: \( 5\vec{w} = 5(-2\hat{\jmath}) = -10\hat{\jmath} \), and \( 2\vec{v} = 2(3\hat{\imath} + 4\hat{\jmath}) = 6\hat{\imath} + 8\hat{\jmath} \). Subtract: \[ -10\hat{\jmath} - (6\hat{\imath} + 8\hat{\jmath}) = -6\hat{\imath} - 18\hat{\jmath} \]. This results in a vector.

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

First, find the unit vector \( \hat{v} = \frac{\vec{v}}{\|\vec{v}\|} = \frac{3\hat{\imath} + 4\hat{\jmath}}{5} = \frac{3}{5}\hat{\imath} + \frac{4}{5}\hat{\jmath} \). Multiply by \( \|\vec{w}\|=2 \): \[ 2(\frac{3}{5}\hat{\imath} + \frac{4}{5}\hat{\jmath}) = \frac{6}{5}\hat{\imath} + \frac{8}{5}\hat{\jmath} \]. This gives a vector.

Step 7: Verify the Parallelogram Law

Calculate each side separately. Left side: \(\|\vec{v}\|^2 + \|\vec{w}\|^2 = 5^2 + 2^2 = 29 \).Calculate \( \|\vec{v}+\vec{w}\|^2 = (\sqrt{13})^2 = 13 \) and \( \|\vec{v}-\vec{w}\|^2 \). Find \( \vec{v}-\vec{w} = 3\hat{\imath} + 6\hat{\jmath} \) and \( \|\vec{v}-\vec{w}\|^2 = \sqrt{3^2 + 6^2}^2 = 45 \).Right side: \(\frac{1}{2}(13 + 45) = 29 \), which equals the left side, verifying the law.

Key Concepts

Vector Addition

Vector addition is a fundamental operation in vector mathematics. When adding vectors, you simply add their corresponding components. For instance, if you have vectors \( \vec{v} = 3\hat{\imath} + 4\hat{\jmath} \) and \( \vec{w} = -2\hat{\jmath} \), you add the \( \hat{\imath} \) components (3 + 0) and the \( \hat{\jmath} \) components (4 + (-2)). This results in a new vector: \( \vec{v} + \vec{w} = 3\hat{\imath} + 2\hat{\jmath} \).

It's crucial to remember that vector addition is commutative, meaning \( \vec{v} + \vec{w} = \vec{w} + \vec{v} \). This property makes it flexible in rearranging vector terms, thus helping in solving complex problems. Ultimately, vector addition combines different directions (components) into a resulting vector that reflects the total direction and magnitude, like navigating an airplane route considering winds.

Magnitude of a Vector

The magnitude of a vector is a measure of its length. It is sometimes referred to as the modulus or norm. To find the magnitude of a vector \( \vec{v} = a\hat{\imath} + b\hat{\jmath} \), use the Pythagorean theorem: \( \|\vec{v}\| = \sqrt{a^2 + b^2} \).

This calculation stems from considering the vector as a point in a Cartesian coordinate system. For instance, given \( \vec{v} = 3\hat{\imath} + 4\hat{\jmath} \), the magnitude would be \( \|\vec{v}\| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5 \).

Understanding magnitude is crucial because it describes the size of the vector independently of its direction. It's particularly important when analyzing forces or speeds, where the strength or velocity (magnitude) is key.

Parallelogram Law

The parallelogram law is a geometric way to derive the resultant of two vectors. When two vectors, say \( \vec{v} \) and \( \vec{w} \), originate from the same point, they can be represented as the adjacent sides of a parallelogram.

To find the resultant vector, which is equivalent to \( \vec{v} + \vec{w} \), draw a diagonal of this parallelogram starting at the same point as \( \vec{v} \) and \( \vec{w} \). Mathematically, the law is verified by showing: \( \|\vec{v}\|^2 + \|\vec{w}\|^2 = \frac{1}{2}(\|\vec{v} + \vec{w}\|^2 + \|\vec{v} - \vec{w}\|^2) \).

This principle is vital because it provides a visual method to add vectors and gives insight into the physics of objects affected by multiple forces, ensuring that the algebraic and geometric interpretations align.

Unit Vector

A unit vector is a vector with a magnitude of 1. It indicates direction only since its length is standardized to 1. To find the unit vector \( \hat{v} \) of a vector \( \vec{v} \), divide the vector by its magnitude: \( \hat{v} = \frac{\vec{v}}{\|\vec{v}\|} \).

For example, if \( \vec{v} = 3\hat{\imath} + 4\hat{\jmath} \) with a magnitude of 5, the unit vector is \( \hat{v} = \frac{3\hat{\imath} + 4\hat{\jmath}}{5} = \frac{3}{5}\hat{\imath} + \frac{4}{5}\hat{\jmath} \).

Unit vectors are essential because they form the backbone of defining direction in vector space. They help decompose other vectors into standard directions, crucial for many applications in physics, engineering, and computer graphics, where precise direction control is required.